Chapter 9 Comparing Quantities

Given:

Marked Price (MP) $= \textsf{₹} 800$

Selling Price (SP) $= \textsf{₹} 680$

To Find:

The rate of discount allowed on the shirt.

Solution:

Discount is calculated as the difference between the Marked Price and the Selling Price.

Discount Amount $= \text{MP} - \text{SP}$

Discount Amount $= \textsf{₹} 800 - \textsf{₹} 680$

Discount Amount $= \textsf{₹} 120$

The discount rate is calculated as the discount amount divided by the Marked Price, expressed as a percentage.

Discount Rate $= \frac{\text{Discount Amount}}{\text{Marked Price}} \times 100\%$

Discount Rate $= \frac{\textsf{₹} 120}{\textsf{₹} 800} \times 100\%$

Discount Rate $= \frac{120}{800} \times 100\%$

Discount Rate $= \frac{12\cancel{0}}{80\cancel{0}} \times 100\%$

Discount Rate $= \frac{12}{80} \times 100\%$

Discount Rate $= \frac{\cancel{12}^{3}}{\cancel{80}_{20}} \times 100\%$

Discount Rate $= \frac{3}{20} \times 100\%$

Discount Rate $= 3 \times \frac{\cancel{100}^{5}}{\cancel{20}_{1}}\%$

Discount Rate $= 3 \times 5\%$

Discount Rate $= 15\%$

Thus, the rate of discount allowed on the shirt is $15\%$.

The correct option is (b) 15%.

Given:

$\frac{7}{3} \%$ of a number is $42$.

To Find:

The number.

Solution:

Let the unknown number be $x$.

According to the problem statement, we can write the equation:

$ \left(\frac{7}{3}\right) \% \text{ of } x = 42 $

We know that percentage means 'out of 100', so $\frac{7}{3} \%$ can be written as $\frac{7/3}{100}$.

The equation becomes:

$ \frac{7/3}{100} \times x = 42 $

$ \frac{7}{3 \times 100} \times x = 42 $

$ \frac{7}{300} \times x = 42 $

To find the value of $x$, we need to isolate $x$. We can do this by multiplying both sides of the equation by the reciprocal of $\frac{7}{300}$, which is $\frac{300}{7}$.

$ x = 42 \times \frac{300}{7} $

Now, we can simplify the expression by cancelling out the common factor of $7$ in the numerator and the denominator.

$ x = \frac{\cancel{42}^{6}}{1} \times \frac{300}{\cancel{7}_{1}} $

$ x = 6 \times 300 $

$ x = 1800 $

The number is $1800$.

The correct option is (c) 1800.

Given:

Cost price of 10 shirts = Selling price of 8 shirts.

To Find:

Whether the transaction results in a profit or loss and its percentage.

Solution:

Let the cost price of one shirt be $\text{CP}$ and the selling price of one shirt be $\text{SP}$.

According to the given information, the cost price of 10 shirts is $10 \times \text{CP}$.

The selling price of 8 shirts is $8 \times \text{SP}$.

We are given that the cost price of 10 shirts is equal to the selling price of 8 shirts.

So, we can write the equation:

$ 10 \times \text{CP} = 8 \times \text{SP} $

We can rearrange this equation to find the relationship between SP and CP:

$ \frac{\text{SP}}{\text{CP}} = \frac{10}{8} $

$ \frac{\text{SP}}{\text{CP}} = \frac{\cancel{10}^{5}}{\cancel{8}_{4}} $

$ \frac{\text{SP}}{\text{CP}} = \frac{5}{4} $

This implies $\text{SP} = \frac{5}{4} \times \text{CP}$.

Since $\frac{5}{4} > 1$, the Selling Price (SP) is greater than the Cost Price (CP). This indicates a profit.

Now, we need to calculate the profit percentage.

Profit = SP - CP

Profit = $\frac{5}{4}\text{CP} - \text{CP}$

Profit = $\left(\frac{5}{4} - 1\right)\text{CP}$

Profit = $\left(\frac{5-4}{4}\right)\text{CP}$

Profit = $\frac{1}{4}\text{CP}$

The profit percentage is calculated on the cost price:

Profit $\% = \frac{\text{Profit}}{\text{CP}} \times 100\%$

Profit $\% = \frac{\frac{1}{4}\text{CP}}{\text{CP}} \times 100\%$

Profit $\% = \frac{1}{4} \times 100\%$

Profit $\% = \frac{\cancel{100}^{25}}{\cancel{4}_{1}}\%$

Profit $\% = 25\%$

The transaction results in a profit of $25\%$.

The correct option is (a) Profit of 25%.

Given:

Principal (P) $= \textsf{₹} 1600$

Annual Interest Rate (R) $= 5\%$ per annum

Compounded half yearly

Time (n) $= 1$ year

To Find:

The amount after one year.

Solution:

Since the interest is compounded half yearly, the interest rate and the time period need to be adjusted accordingly.

The half-yearly interest rate $(r)$ is half of the annual rate:

$ r = \frac{\text{Annual Rate}}{2} = \frac{5\%}{2} = 2.5\% $

The number of compounding periods in one year $(t)$ is the number of half-years in one year:

$ t = \text{Time in years} \times 2 = 1 \text{ year} \times 2 \text{ half-years/year} = 2 \text{ periods} $

The formula for the amount (A) when interest is compounded is:

$ A = P\left(1 + \frac{r}{100}\right)^t $

Where P is the principal, r is the interest rate per compounding period (as a percentage), and t is the total number of compounding periods.

Substitute the given values into the formula:

$ A = 1600 \left(1 + \frac{2.5}{100}\right)^2 $

Simplify the term inside the parenthesis:

$ 1 + \frac{2.5}{100} = 1 + 0.025 = 1.025 $

Alternatively, as a fraction:

$ 1 + \frac{2.5}{100} = 1 + \frac{25/10}{100} = 1 + \frac{25}{1000} = 1 + \frac{1}{40} = \frac{40}{40} + \frac{1}{40} = \frac{41}{40} $

Using the fraction form for calculation:

$ A = 1600 \left(\frac{41}{40}\right)^2 $

$ A = 1600 \times \left(\frac{41}{40} \times \frac{41}{40}\right) $

$ A = 1600 \times \frac{41 \times 41}{40 \times 40} $

$ A = 1600 \times \frac{1681}{1600} $

Now, cancel the 1600 from the numerator and the denominator:

$ A = \cancel{1600} \times \frac{1681}{\cancel{1600}} $

$ A = 1681 $

The amount after one year is $\textsf{₹} 1681$.

The correct option is (c) Rs 1681.

Given:

Number of pens sold $= 50$

Loss on selling 50 pens = Selling Price of 10 pens

To Find:

The loss per cent.

Solution:

Let the Cost Price (CP) of one pen be $C$.

Let the Selling Price (SP) of one pen be $S$.

The Cost Price of 50 pens $= 50 \times C = 50C$.

The Selling Price of 50 pens $= 50 \times S = 50S$.

The Loss amount is given as the selling price of 10 pens.

Loss $= 10 \times S = 10S$.

We know that Loss = Cost Price - Selling Price.

For the transaction of 50 pens, the total loss is the total cost price minus the total selling price.

Total Loss on 50 pens = CP of 50 pens - SP of 50 pens

Substitute the values we have:

$ 10S = 50C - 50S $

Now, rearrange the equation to find the relationship between $C$ and $S$. Add $50S$ to both sides:

$ 10S + 50S = 50C $

$ 60S = 50C $

We can express $50C$ in terms of $S$. This is useful for calculating the loss percentage, which is defined as $\frac{\text{Loss}}{\text{CP}} \times 100\%$. The CP in the formula is the cost price of the items sold, which is $50C$.

Loss Per cent $= \frac{\text{Loss}}{\text{CP of 50 pens}} \times 100\% $

Substitute the values: Loss $= 10S$ and CP of 50 pens $= 50C$.

Loss Per cent $= \frac{10S}{50C} \times 100\% $

From the relation $60S = 50C$, we can substitute $50C$ with $60S$ in the denominator:

Loss Per cent $= \frac{10S}{60S} \times 100\% $

Cancel out the $S$ and simplify the fraction:

Loss Per cent $= \frac{\cancel{10S}}{\cancel{60S}} \times 100\% $

Loss Per cent $= \frac{10}{60} \times 100\% $

Loss Per cent $= \frac{\cancel{10}^{1}}{\cancel{60}_{6}} \times 100\% $

Loss Per cent $= \frac{1}{6} \times 100\% $

Loss Per cent $= \frac{100}{6}\% $

Loss Per cent $= \frac{\cancel{100}^{50}}{\cancel{6}_{3}}\% $

Loss Per cent $= \frac{50}{3}\% $

As a mixed fraction, $\frac{50}{3} = 16$ with a remainder of $2$, so $\frac{50}{3}\% = 16\frac{2}{3}\%$.

His loss per cent is $16\frac{2}{3}\%$.

The discount per cent is calculated on the marked price of an article.

Explanation:

The discount offered on an article is always a reduction from its initial or advertised price. This initial price is known as the Marked Price or List Price.

When a shopkeeper offers a discount, they are reducing the price from what is written on the tag or label (the Marked Price) to arrive at the Selling Price.

Therefore, the discount amount is the difference between the Marked Price and the Selling Price.

The discount percentage is calculated relative to the price from which the discount is being subtracted, which is the Marked Price.

The formula for Discount Percentage is:

$ \text{Discount} \% = \frac{\text{Discount Amount}}{\text{Marked Price}} \times 100 $

Calculating the discount percentage based on the Selling Price or Cost Price would not represent the reduction offered to the customer from the advertised price.

Given:

Total price of the toy including sales tax $= \textsf{₹} 660$

Sales tax rate $= 10\%$

To Find:

The selling price of the toy (price before sales tax).

Solution:

Let the selling price of the toy be $x$. The sales tax is calculated on this selling price.

Sales Tax Amount $= 10\%$ of the selling price

Sales Tax Amount $= 10\%$ of $x$

Sales Tax Amount $= \frac{10}{100} \times x = 0.10x$

The total price paid is the sum of the selling price and the sales tax amount.

Total Price $= \text{Selling Price} + \text{Sales Tax Amount}

We are given that the Total Price is $\textsf{₹} 660$.

$ 660 = x + 0.10x $

Combine the terms involving $x$:

$ 660 = (1 + 0.10)x $

$ 660 = 1.10x $

To find $x$, divide both sides of the equation by $1.10$:

$ x = \frac{660}{1.10} $

To remove the decimal from the denominator, multiply the numerator and denominator by 10:

$ x = \frac{660 \times 10}{1.10 \times 10} $

$ x = \frac{6600}{11} $

Now, perform the division:

$ x = 600 $

So, the selling price of the toy is $\textsf{₹} 600$.

The selling price of the toy is $\textsf{₹} 600$.

False (F)

Explanation:

When interest is compounded half yearly, it means that the interest is calculated and added to the principal twice a year (every six months).

A year consists of 12 months.

A half-year is equal to $\frac{12 \text{ months}}{2} = 6$ months.

So, in one year, there are $\frac{12 \text{ months}}{6 \text{ months/period}} = 2$ half-year periods.

Therefore, the number of conversion periods in a year when interest is compounded half yearly is 2, not 4.

If the interest were compounded quarterly, then there would be 4 conversion periods in a year (every three months).

True (T)

Explanation:

The cost of the book before sales tax is $\textsf{₹} 600$.

The rate of sales tax is $7\%$.

Sales tax is calculated on the original cost of the item.

Sales Tax Amount $= 7\%$ of $\textsf{₹} 600$

Sales Tax Amount $= \frac{7}{100} \times 600$

Sales Tax Amount $= \frac{7}{\cancel{100}} \times \cancel{600}^{6}$

Sales Tax Amount $= 7 \times 6$

Sales Tax Amount $= \textsf{₹} 42$

The total amount payable is the cost of the book plus the sales tax amount.

Total Amount Payable $= \text{Cost of book} + \text{Sales Tax Amount}

Total Amount Payable $= \textsf{₹} 600 + \textsf{₹} 42$

Total Amount Payable $= \textsf{₹} 642$

The calculated total amount payable is $\textsf{₹} 642$, which matches the amount stated in the question.

Therefore, the statement is true.

True (T)

Explanation:

Given:

Selling Price (SP) $= \textsf{₹} 680$

Discount Rate $= 15\%$

Stated Marked Price (MP) $= \textsf{₹} 800$

To Verify:

Whether the actual marked price is $\textsf{₹} 800$ based on the given SP and discount rate.

Solution:

We know that the Selling Price is calculated by subtracting the Discount Amount from the Marked Price.

$ \text{SP} = \text{MP} - \text{Discount Amount} $

The Discount Amount is the Discount Rate applied to the Marked Price.

$ \text{Discount Amount} = \text{Discount Rate} \times \text{MP} $

$ \text{Discount Amount} = 15\% \text{ of MP} $

$ \text{Discount Amount} = \frac{15}{100} \times \text{MP} = 0.15 \times \text{MP} $

Substitute this into the SP formula:

$ \text{SP} = \text{MP} - 0.15 \times \text{MP} $

$ \text{SP} = (1 - 0.15) \times \text{MP} $

$ \text{SP} = 0.85 \times \text{MP} $

We are given that SP $= \textsf{₹} 680$. So,

$ 680 = 0.85 \times \text{MP} $

To find the Marked Price (MP), divide the Selling Price by $0.85$:

$ \text{MP} = \frac{680}{0.85} $

To simplify the division, multiply the numerator and the denominator by 100:

$ \text{MP} = \frac{680 \times 100}{0.85 \times 100} = \frac{68000}{85} $

Now, divide 68000 by 85.

$ \frac{68000}{85} = \frac{680 \times 100}{85} $

We can simplify $\frac{680}{85}$. Both are divisible by 5:

$ 680 \div 5 = 136 $

$ 85 \div 5 = 17 $

So, $ \text{MP} = \frac{136}{17} \times 100 $

Now, simplify $\frac{136}{17}$. We know that $17 \times 8 = 136$.

$ \text{MP} = 8 \times 100 $

$ \text{MP} = 800 $

The calculated Marked Price is $\textsf{₹} 800$. This matches the marked price given in the statement.

Therefore, the statement is true.

True (T)

Explanation:

In business and accounting, the Cost Price of an item or a product usually refers to the total expenditure incurred by a seller to acquire or produce that item. This cost often includes not only the direct cost of purchasing or manufacturing the item (like the price paid to a supplier or the cost of raw materials and labour) but also additional costs necessary to make the item available for sale or to run the business related to that item.

These additional costs are called overhead charges or indirect costs. Examples include:

• Transportation or freight charges to bring the goods to the shop/warehouse.
• Loading and unloading costs.
• Insurance costs.
• Storage costs.
• Packaging costs (sometimes).
• Other related expenses incurred before the sale.

Including these overheads in the cost price helps in determining the true cost of the item to the seller, which is essential for calculating the actual profit or loss when the item is sold.

So, yes, overhead charges are often included in the cost price to arrive at a more accurate figure of the total cost incurred by the seller.

Given:

A number is first increased by $20\%$.

The resulting number is then decreased by $20\%$.

To Find:

The net increase or decrease per cent.

Solution:

Let the original number be $x$.

When the number is increased by $20\%$, the new number is:

$ \text{New Number}_1 = x + 20\% \text{ of } x $

$ \text{New Number}_1 = x + \frac{20}{100} \times x $

$ \text{New Number}_1 = x + 0.20x $

$ \text{New Number}_1 = 1.20x $

Now, this new number ($1.20x$) is decreased by $20\%$. The decrease amount is $20\%$ of $1.20x$.

$ \text{Decrease Amount} = 20\% \text{ of } 1.20x $

$ \text{Decrease Amount} = \frac{20}{100} \times 1.20x $

$ \text{Decrease Amount} = 0.20 \times 1.20x $

$ \text{Decrease Amount} = 0.24x $

The final number is obtained by subtracting the Decrease Amount from $\text{New Number}_1$.

$ \text{Final Number} = \text{New Number}_1 - \text{Decrease Amount} $

$ \text{Final Number} = 1.20x - 0.24x $

$ \text{Final Number} = (1.20 - 0.24)x $

$ \text{Final Number} = 0.96x $

To find the net change, we compare the Final Number with the Original Number.

$ \text{Net Change} = \text{Final Number} - \text{Original Number} $

$ \text{Net Change} = 0.96x - x $

$ \text{Net Change} = (0.96 - 1)x $

$ \text{Net Change} = -0.04x $

Since the net change is negative ($-0.04x$), there is a net decrease.

The amount of net decrease is $0.04x$.

To find the net decrease per cent, we calculate the net change as a percentage of the Original Number.

$ \text{Net Decrease Per cent} = \frac{\text{Net Decrease}}{\text{Original Number}} \times 100\% $

$ \text{Net Decrease Per cent} = \frac{0.04x}{x} \times 100\% $

$ \text{Net Decrease Per cent} = 0.04 \times 100\% $

$ \text{Net Decrease Per cent} = 4\% $

The net result is a decrease of $4\%$.

Alternate Solution (Using 100):

Assume the original number is $100$.

Increase by $20\%$: New number $= 100 + 20\% \text{ of } 100 = 100 + 20 = 120$.

Decrease the new number ($120$) by $20\%$: Decrease amount $= 20\% \text{ of } 120 = \frac{20}{100} \times 120 = \frac{1}{5} \times 120 = 24$.

Final number $= 120 - 24 = 96$.

Net change $= \text{Final Number} - \text{Original Number} = 96 - 100 = -4$.

Since the original number was $100$, the net change is $-4\%$. This means a $4\%$ decrease.

Conclusion:

There is a net decrease of $4\%$.

Note: When a number is increased by $x\%$ and then decreased by the same $x\%$, there is always a net decrease. The net decrease percentage is given by the formula $\left(\frac{x}{10}\right)^2\%$. In this case, $x=20$, so the decrease is $\left(\frac{20}{10}\right)^2\% = 2^2\% = 4\%$.

Given:

Marked Price (MP) $= \textsf{₹} 280$

Discount Rate $= 20\%$

Profit Rate $= 12\%$

To Find:

The Cost Price (CP) of the article.

Solution:

First, we need to find the Selling Price (SP) of the article after the discount is applied to the Marked Price.

Discount Amount $= \text{Discount Rate} \times \text{MP}

Discount Amount $= 20\% \text{ of } \textsf{₹} 280$

Discount Amount $= \frac{20}{100} \times 280$

Discount Amount $= 0.20 \times 280$

Discount Amount $= \textsf{₹} 56$

The Selling Price is the Marked Price minus the Discount Amount.

$ \text{SP} = \text{MP} - \text{Discount Amount} $

$ \text{SP} = \textsf{₹} 280 - \textsf{₹} 56 $

$ \text{SP} = \textsf{₹} 224 $

Now, we know the Selling Price and the Profit Rate. The profit is calculated on the Cost Price.

Profit $= \text{Profit Rate} \times \text{CP}

Profit $= 12\% \text{ of CP}

Profit $= \frac{12}{100} \times \text{CP} = 0.12 \times \text{CP}$

The Selling Price is also the Cost Price plus the Profit.

$ \text{SP} = \text{CP} + \text{Profit} $

Substitute the expressions for SP and Profit:

$ 224 = \text{CP} + 0.12 \times \text{CP} $

$ 224 = (1 + 0.12) \times \text{CP} $

$ 224 = 1.12 \times \text{CP} $

To find the Cost Price (CP), divide the Selling Price by $1.12$.

$ \text{CP} = \frac{224}{1.12} $

To perform the division, we can remove the decimal by multiplying the numerator and denominator by 100.

$ \text{CP} = \frac{224 \times 100}{1.12 \times 100} $

$ \text{CP} = \frac{22400}{112} $

Divide 22400 by 112:

$ \frac{22400}{112} = \frac{\cancel{224}^{2} \times 100}{\cancel{112}^{1}} $

$ \text{CP} = 2 \times 100 $

$ \text{CP} = 200 $

The Cost Price of the article is $\textsf{₹} 200$.

Given:

Principal (P) $= \textsf{₹} 48,000$

Annual Interest Rate (R) $= 8\%$ per annum

Time (n) $= 1$ year

Compounding Frequency: Half yearly

To Find:

The Compound Interest (CI).

Solution:

When interest is compounded half yearly, the annual interest rate is halved, and the time period is doubled in terms of compounding periods.

Half-yearly interest rate (r) $= \frac{\text{Annual Rate}}{2} = \frac{8\%}{2} = 4\% $ per half-year.

Number of compounding periods (t) in 1 year $= 1 \text{ year} \times 2 \text{ half-years/year} = 2$ periods.

The formula for the Amount (A) when compounded half yearly is:

$ A = P\left(1 + \frac{r}{100}\right)^t $

Substitute the given values:

$ A = 48000\left(1 + \frac{4}{100}\right)^2 $

$ A = 48000\left(1 + 0.04\right)^2 $

$ A = 48000\left(1.04\right)^2 $

Calculate $(1.04)^2$:

$ (1.04)^2 = 1.04 \times 1.04 = 1.0816 $

Now, calculate the Amount:

$ A = 48000 \times 1.0816 $

$ A = 48 \times 1000 \times 1.0816 $

$ A = 48 \times 1081.6 $

Let's perform the multiplication:

$ A = \textsf{₹} 51916.80 $

The Compound Interest (CI) is the difference between the Amount and the Principal.

$ \text{CI} = A - P $

$ \text{CI} = \textsf{₹} 51916.80 - \textsf{₹} 48000 $

$ \text{CI} = \textsf{₹} 3916.80 $

The compound interest is $\textsf{₹} 3916.80$.

Given:

Cost price of 1 dozen (12) lemons $= \textsf{₹} 60$

Selling price of 10 lemons $= \textsf{₹} 40$

To Find:

The gain or loss percent.

Solution:

To compare the cost and selling prices, we need to find the price per unit (per lemon) or the price for the same number of lemons in both cases.

Let's find the cost price and selling price of one lemon.

Cost Price (CP) of 1 dozen (12) lemons $= \textsf{₹} 60$

CP of 1 lemon $= \frac{\text{Total CP}}{\text{Number of lemons}}$

CP of 1 lemon $= \frac{60}{12}$

CP of 1 lemon $= \textsf{₹} 5$

Selling Price (SP) of 10 lemons $= \textsf{₹} 40$

SP of 1 lemon $= \frac{\text{Total SP}}{\text{Number of lemons}}$

SP of 1 lemon $= \frac{40}{10}$

SP of 1 lemon $= \textsf{₹} 4$

Compare the CP and SP of one lemon:

CP per lemon $= \textsf{₹} 5$

SP per lemon $= \textsf{₹} 4$

Since SP $<$ CP, there is a loss in the transaction.

Loss per lemon $= \text{CP} - \text{SP}$

Loss per lemon $= \textsf{₹} 5 - \textsf{₹} 4$

Loss per lemon $= \textsf{₹} 1$

Now, calculate the loss percentage. The loss percentage is always calculated on the cost price.

Loss $\% = \frac{\text{Loss}}{\text{CP}} \times 100\%$

Loss $\% = \frac{\text{Loss per lemon}}{\text{CP per lemon}} \times 100\%$

Loss $\% = \frac{1}{5} \times 100\%$

Loss $\% = \frac{\cancel{100}^{20}}{\cancel{5}_{1}}\%$

Loss $\% = 20\%$

The transaction results in a loss of $20\%$.

Alternate Solution (Using LCM):

Find the LCM of the number of items bought (12) and the number of items sold (10).

LCM$(12, 10) = 60$

Calculate the CP and SP for 60 lemons.

CP of 12 lemons $= \textsf{₹} 60$

CP of 1 lemon $= \textsf{₹} \frac{60}{12} = \textsf{₹} 5$

CP of 60 lemons $= 60 \times \textsf{₹} 5 = \textsf{₹} 300$

SP of 10 lemons $= \textsf{₹} 40$

SP of 1 lemon $= \textsf{₹} \frac{40}{10} = \textsf{₹} 4$

SP of 60 lemons $= 60 \times \textsf{₹} 4 = \textsf{₹} 240$

Compare the total CP and SP for 60 lemons:

CP $= \textsf{₹} 300$

SP $= \textsf{₹} 240$

Since SP $<$ CP, there is a loss.

Loss $= \text{CP} - \text{SP}$

Loss $= \textsf{₹} 300 - \textsf{₹} 240$

Loss $= \textsf{₹} 60$

Loss $\% = \frac{\text{Loss}}{\text{CP}} \times 100\%$

Loss $\% = \frac{60}{300} \times 100\%$

Loss $\% = \frac{60}{300} \times 100\%$

Loss $\% = \frac{\cancel{60}^{1}}{\cancel{300}_{5}} \times 100\%$

Loss $\% = \frac{1}{5} \times 100\%$

Loss $\% = 20\%$

Both methods yield the same result. The net result is a loss of $20\%$.

Solution:

Let P be the principal amount, R be the annual interest rate, and T be the time in years.

The formula for Simple Interest (S) is:

$ S = \frac{P \times R \times T}{100} $

The formula for the Amount (A) under Compound Interest compounded annually is:

$ A = P \left(1 + \frac{R}{100}\right)^T $

The Compound Interest (C) is given by $ C = A - P $:

$ C = P \left(1 + \frac{R}{100}\right)^T - P $

Let's compare S and C for different values of T (assuming T is in years and compounding is annual):

Case 1: Time T = 1 year

Simple Interest: $ S = \frac{P \times R \times 1}{100} = \frac{PR}{100} $

Compound Interest: $ A = P \left(1 + \frac{R}{100}\right)^1 = P \left(1 + \frac{R}{100}\right) = P + \frac{PR}{100} $

$ C = A - P = \left(P + \frac{PR}{100}\right) - P = \frac{PR}{100} $

For T = 1 year, we have $ S = C $. This means possibility (ii) is correct.

Case 2: Time T > 1 year

When T is greater than 1 year, compound interest calculates interest on the principal amount as well as on the accumulated interest from previous periods. Simple interest, on the other hand, calculates interest only on the original principal amount for the entire period.

Mathematically, for $T > 1$ and $R > 0$, the inequality $ \left(1 + \frac{R}{100}\right)^T > \left(1 + \frac{R}{100} \times T\right) $ holds true.

Multiplying by P (assuming P > 0):

$ P \left(1 + \frac{R}{100}\right)^T > P \left(1 + \frac{RT}{100}\right) $

$ P \left(1 + \frac{R}{100}\right)^T > P + \frac{PRT}{100} $

$ P \left(1 + \frac{R}{100}\right)^T - P > \frac{PRT}{100} $

This shows that $ C > S $ for $ T > 1 $ year. This means possibility (i) is correct.

Case 3: Time 0 < T < 1 year

If the time period T is less than 1 year, and assuming compounding is still annual, the standard formula for compound interest is used with a fractional exponent T. For $0 < T < 1$ and $R > 0$, the inequality $ \left(1 + \frac{R}{100}\right)^T < \left(1 + \frac{R}{100} \times T\right) $ holds true (based on Bernoulli's inequality).

Multiplying by P:

$ P \left(1 + \frac{R}{100}\right)^T < P \left(1 + \frac{RT}{100}\right) $

$ P \left(1 + \frac{R}{100}\right)^T < P + \frac{PRT}{100} $

$ P \left(1 + \frac{R}{100}\right)^T - P < \frac{PRT}{100} $

This shows that $ C < S $ for $ 0 < T < 1 $ year (with annual compounding). This means possibility (iii) is correct.

Based on the analysis, if T can be any positive value, then all three possibilities (i) C > S, (ii) C = S, and (iii) C < S are possible depending on whether T > 1, T = 1, or 0 < T < 1.

However, in the standard context where compound interest is compared with simple interest, the comparison is usually made for a time period of one year or more, as it takes at least one compounding period for the effect of compounding (earning interest on interest) to manifest.

If we consider the standard scenario where T $\geq 1$ year and compounding is annual:

• If T = 1 year, $ C = S $ (possibility (ii)).
• If T > 1 year, $ C > S $ (possibility (i)).

In this common interpretation, either C > S or C = S is correct.

Considering the given options, the most appropriate answer based on the standard comparison of SI and CI for periods of one year or more is that either (i) C > S or (ii) C = S is correct.

The correct option is (b) either (i) or (ii) is correct.

Given:

A sum (Principal P) doubles in 2 years.

Scenario 1: Simple Interest at r% per annum.

Scenario 2: Compound Interest compounded annually at R% per annum.

To Compare:

The rates r and R.

Solution:

Let the principal amount be $P$.

When the sum doubles, the amount becomes $2P$.

Case 1: Simple Interest

Principal $= P$

Amount $= 2P$

Interest earned $= \text{Amount} - \text{Principal} = 2P - P = P$

Time $(T) = 2$ years

Rate $= r\%$ per annum

The formula for Simple Interest is $ \text{SI} = \frac{P \times r \times T}{100} $. Substituting the values:

$ P = \frac{P \times r \times 2}{100} $

Assuming $P \neq 0$, we can divide both sides by $P$:

$ 1 = \frac{2r}{100} $

Multiply both sides by 100:

$ 100 = 2r $

Divide by 2:

$ r = \frac{100}{2} = 50 $

So, the simple interest rate $r = 50\%$.

Case 2: Compound Interest (Compounded Annually)

Principal $= P$

Amount $= 2P$

Time $(T) = 2$ years

Rate $= R\%$ per annum

The formula for Amount under Compound Interest is $ A = P\left(1 + \frac{R}{100}\right)^T $.

Substitute the values:

$ 2P = P\left(1 + \frac{R}{100}\right)^2 $

Assuming $P \neq 0$, divide both sides by $P$:

$ 2 = \left(1 + \frac{R}{100}\right)^2 $

Take the square root of both sides (since rate R must be positive):

$ \sqrt{2} = 1 + \frac{R}{100} $

Subtract 1 from both sides:

$ \sqrt{2} - 1 = \frac{R}{100} $

Multiply by 100:

$ R = 100(\sqrt{2} - 1) $

We know that $\sqrt{2} \approx 1.4142$.

$ R \approx 100(1.4142 - 1) $

$ R \approx 100(0.4142) $

$ R \approx 41.42 $

So, the compound interest rate $R \approx 41.42\%$.

Comparison of r and R:

We found $r = 50\%$ and $R \approx 41.42\%$.

Comparing these values, we see that $R$ is less than $r$.

$ R < r $

Therefore, the correct relationship is $R < r$.

The correct option is (b) R < r.

Given:

Principal (P) $= \textsf{₹} 50,000$

Annual Interest Rate (R) $= 4\%$ per annum

Time (n) $= 2$ years

Compounding Frequency: Annually

To Find:

The Compound Interest (CI).

Solution:

The formula for the Amount (A) when compounded annually is:

$ A = P\left(1 + \frac{R}{100}\right)^n $

Substitute the given values:

$ A = 50000\left(1 + \frac{4}{100}\right)^2 $

$ A = 50000\left(1 + 0.04\right)^2 $

$ A = 50000\left(1.04\right)^2 $

Calculate $(1.04)^2$:

$ (1.04)^2 = 1.04 \times 1.04 $

$ (1.04)^2 = 1.0816 $

Now, calculate the Amount:

$ A = 50000 \times 1.0816 $

$ A = 5 \times 10000 \times 1.0816 $

$ A = 5 \times 10816 $

$ A = \textsf{₹} 54080 $

The Compound Interest (CI) is the difference between the Amount and the Principal.

$ \text{CI} = A - P $

$ \text{CI} = \textsf{₹} 54080 - \textsf{₹} 50000 $

$ \text{CI} = \textsf{₹} 4080 $

The compound interest is $\textsf{₹} 4080$.

The correct option is (b) Rs 4,080.

Given:

Marked Price (MP) $= \textsf{₹} 1,200$

Discount Rate $= 12\%$

To Find:

The Selling Price (SP) of the article.

Solution:

The Discount Amount is calculated on the Marked Price.

Discount Amount $= \text{Discount Rate} \times \text{MP}

Discount Amount $= 12\% \text{ of } \textsf{₹} 1,200$

Discount Amount $= \frac{12}{100} \times 1200$

Discount Amount $= 0.12 \times 1200$

Discount Amount $= \textsf{₹} 144$

The Selling Price is the Marked Price minus the Discount Amount.

$ \text{SP} = \text{MP} - \text{Discount Amount} $

$ \text{SP} = \textsf{₹} 1200 - \textsf{₹} 144 $

$ \text{SP} = \textsf{₹} 1056 $

Alternate Method:

If a discount of $12\%$ is given on the Marked Price, the Selling Price is the remaining percentage of the Marked Price.

Percentage paid $= 100\% - \text{Discount Rate} = 100\% - 12\% = 88\%$

Selling Price (SP) $= 88\%$ of Marked Price (MP)

$ \text{SP} = \frac{88}{100} \times 1200 $

$ \text{SP} = 0.88 \times 1200 $

$ \text{SP} = 88 \times 12 $

$ \text{SP} = \textsf{₹} 1056 $

The selling price of the article is $\textsf{₹} 1056$.

The correct option is (a) Rs 1,056.

Given:

$90\%$ of $x$ is $315$ km.

To Find:

The value of $x$.

Solution:

According to the problem statement, we can translate the given information into an equation:

$ 90\% \text{ of } x = 315 \text{ km} $

Recall that a percentage can be written as a fraction by dividing by 100.

$ \frac{90}{100} \times x = 315 $

Simplify the fraction $\frac{90}{100}$:

$ \frac{9}{10} \times x = 315 $

To find the value of $x$, we need to isolate $x$. Multiply both sides of the equation by the reciprocal of $\frac{9}{10}$, which is $\frac{10}{9}$.

$ x = 315 \times \frac{10}{9} $

Now, we can simplify by dividing 315 by 9.

$ 315 \div 9 = \frac{315}{9} $

Performing the division:

$ 31 \div 9 = 3 $ with a remainder of $31 - (9 \times 3) = 31 - 27 = 4$.

Bring down the 5, making it 45.

$ 45 \div 9 = 5 $. No remainder.

So, $\frac{315}{9} = 35$.

Substitute this back into the equation for $x$:

$ x = 35 \times 10 $

$ x = 350 $

The unit given is kilometers (km), so the value of $x$ is $350$ km.

The value of $x$ is $350$ km.

The correct option is (b) 350 km.

Given:

Cost Price (CP) $= \textsf{₹} 360$

Desired Profit Rate $= 25\%$

Allowed Discount Rate $= 10\%$

To Find:

The Marked Price (MP) of the article.

Solution:

First, we need to determine the Selling Price (SP) at which the article must be sold to gain a $25\%$ profit on the Cost Price.

Profit $= \text{Profit Rate} \times \text{CP}

Profit $= 25\% \text{ of } \textsf{₹} 360$

Profit $= \frac{25}{100} \times 360$

Profit $= \frac{1}{4} \times 360$

Profit $= \textsf{₹} 90$

The Selling Price is the Cost Price plus the Profit.

$ \text{SP} = \text{CP} + \text{Profit} $

$ \text{SP} = \textsf{₹} 360 + \textsf{₹} 90 $

$ \text{SP} = \textsf{₹} 450 $

Alternatively, SP can be directly calculated as $(100\% + \text{Profit Rate})$ of CP:

$ \text{SP} = (100\% + 25\%) \text{ of CP} = 125\% \text{ of } \textsf{₹} 360$

$ \text{SP} = \frac{125}{100} \times 360 = 1.25 \times 360 $

$ \text{SP} = \textsf{₹} 450 $

Now, we know the Selling Price ($\textsf{₹} 450$) and the Discount Rate ($10\%$). The discount is applied on the Marked Price (MP) to arrive at the Selling Price.

This means the Selling Price is $(100\% - \text{Discount Rate})$ of the Marked Price.

$ \text{SP} = (100\% - 10\%) \text{ of MP} $

$ \text{SP} = 90\% \text{ of MP} $

$ \textsf{₹} 450 = \frac{90}{100} \times \text{MP} $

$ 450 = 0.90 \times \text{MP} $

To find the Marked Price (MP), divide the Selling Price by $0.90$:

$ \text{MP} = \frac{450}{0.90} $

To remove the decimal, multiply the numerator and denominator by 100:

$ \text{MP} = \frac{450 \times 100}{0.90 \times 100} = \frac{45000}{90} $

Simplify the fraction by cancelling out a factor of 10 from numerator and denominator, and then dividing by 9:

$ \text{MP} = \frac{4500\cancel{0}}{9\cancel{0}} = \frac{4500}{9} $

$ \text{MP} = 500 $

The Marked Price of the article must be $\textsf{₹} 500$.

The shopkeeper must mark the price of the article at $\textsf{₹} 500$.

The correct option is (a) Rs 500.

Given:

Marked Price (MP) $= x$

Discount Rate $= a\%$

To Find:

The Discount Amount.

Solution:

The discount is a percentage of the Marked Price.

Discount Amount $= \text{Discount Rate} \times \text{Marked Price}

Discount Amount $= a\% \text{ of } x$

Recall that a percentage $a\%$ means $\frac{a}{100}$.

Discount Amount $= \frac{a}{100} \times x$

Discount Amount $= x \times \frac{a}{100}$

Comparing this expression with the given options:

(a) $\frac{x}{a} \times 100$: This is incorrect.

(b) $\frac{a}{x} \times 100$: This is the formula for discount percentage, not the discount amount.

(c) $x \times \frac{a}{100}$: This matches our calculated discount amount.

(d) $\frac{100}{x \times a}$: This is incorrect.

The discount is $x \times \frac{a}{100}$.

The correct option is (c) x × $\frac{a}{100}$.

Given:

Principal (P) $= \textsf{₹} 1,00,000$

Annual Interest Rate (R) $= 12\%$ per annum

Compounding Frequency: Half yearly

Amount paid (A) $= \textsf{₹} 1,12,360$

Identity: $(1.06)^2 = 1.1236$

To Find:

The period (time) for which the loan was taken.

Solution:

When interest is compounded half yearly, the interest rate is adjusted per half-year, and the time is counted in half-year periods.

Half-yearly interest rate (r) $= \frac{\text{Annual Rate}}{2} = \frac{12\%}{2} = 6\% $ per half-year.

Let the number of half-yearly compounding periods be $t$.

The formula for the Amount (A) when compounded half yearly is:

$ A = P\left(1 + \frac{r}{100}\right)^t $

Substitute the given values (using the half-yearly rate r = 6):

$ 112360 = 100000\left(1 + \frac{6}{100}\right)^t $

$ 112360 = 100000\left(1.06\right)^t $

Divide both sides by 100000:

$ \frac{112360}{100000} = \left(1.06\right)^t $

$ 1.1236 = \left(1.06\right)^t $

We are given that $(1.06)^2 = 1.1236$.

So, we can substitute this into the equation:

$ (1.06)^2 = \left(1.06\right)^t $

Since the bases are equal, the exponents must be equal.

$ t = 2 $

Here, $t$ represents the number of half-yearly periods.

Total Time in years $= \frac{\text{Number of half-yearly periods}}{2}$

Total Time $= \frac{t}{2} \text{ years}$

Total Time $= \frac{2}{2} \text{ years}$

Total Time $= 1 \text{ year}$

The period for which she took the loan is 1 year.

The correct option is (b) 1 year.

Solution:

Let the principal be $P$, the given annual rate be $R\%$, and the given time be $T$ years.

When interest is compounded half yearly, the interest is calculated and added to the principal twice a year, i.e., every six months.

The interest rate per compounding period is half of the annual rate because a half-year is $\frac{1}{2}$ of a year.

Rate per half-year $= \frac{\text{Annual Rate}}{2} = \frac{R}{2}\% $

So, the given annual rate is halved.

The number of compounding periods in the given time is found by counting how many half-years are there in the total time period $T$ years.

Number of half-year periods $= \text{Time in years} \times 2 $

Number of half-year periods $= T \times 2 = 2T $ periods.

So, the given number of years is doubled to get the number of compounding periods.

Therefore, for calculation of interest compounded half yearly, we use half the given annual rate and double the given number of years as the number of compounding periods.

Comparing this with the given options:

• (a) Double the given annual rate and half the given number of years. (Incorrect)
• (b) Double the given annual rate as well as the given number of years. (Incorrect)
• (c) Half the given annual rate as well as the given number of years. (Incorrect)
• (d) Half the given annual rate and double the given number of years. (Correct)

The correct option is (d) Half the given annual rate and double the given number of years.

Given:

Cost of the scooter $= \textsf{₹} 36,450$

Sales Tax Rate $= 9\%$

To Find:

The total amount paid by Shyama.

Solution:

The sales tax is calculated as a percentage of the cost of the scooter.

Sales Tax Amount $= \text{Sales Tax Rate} \times \text{Cost of Scooter}

Sales Tax Amount $= 9\% \text{ of } \textsf{₹} 36,450$

Sales Tax Amount $= \frac{9}{100} \times 36450$

Sales Tax Amount $= 0.09 \times 36450$

Let's perform the multiplication:

$ 36450 \times 0.09 $

Since there are two decimal places in 0.09, we place the decimal point two places from the right in the result.

Sales Tax Amount $= \textsf{₹} 3280.50$

The total amount paid is the sum of the cost of the scooter and the sales tax amount.

Total Amount Paid $= \text{Cost of Scooter} + \text{Sales Tax Amount}

Total Amount Paid $= \textsf{₹} 36450 + \textsf{₹} 3280.50$

Total Amount Paid $= \textsf{₹} 39730.50$

The total amount paid by Shyama is $\textsf{₹} 39730.50$.

The correct option is (b) Rs 39,730.50.

Given:

Marked Price (MP) $= \textsf{₹} 80$

Selling Price (SP) $= \textsf{₹} 76$

To Find:

The discount rate.

Solution:

The Discount Amount is the difference between the Marked Price and the Selling Price.

$ \text{Discount Amount} = \text{MP} - \text{SP} $

$ \text{Discount Amount} = \textsf{₹} 80 - \textsf{₹} 76 $

$ \text{Discount Amount} = \textsf{₹} 4$

The Discount Rate is calculated as the Discount Amount divided by the Marked Price, expressed as a percentage.

$ \text{Discount Rate } \% = \frac{\text{Discount Amount}}{\text{Marked Price}} \times 100\% $

$ \text{Discount Rate } \% = \frac{\textsf{₹} 4}{\textsf{₹} 80} \times 100\% $

$ \text{Discount Rate } \% = \frac{4}{80} \times 100\% $

Simplify the fraction $\frac{4}{80}$:

$ \frac{4}{80} = \frac{\cancel{4}^{1}}{\cancel{80}_{20}} = \frac{1}{20} $

Now, calculate the percentage:

$ \text{Discount Rate } \% = \frac{1}{20} \times 100\% $

$ \text{Discount Rate } \% = \frac{100}{20}\% $

$ \text{Discount Rate } \% = 5\% $

The discount rate is $5\%$.

The correct option is (a) 5%.

Given:

Cost price for A (CP$_A$) $= \textsf{₹} 8,000$

Profit percentage for A $= 20\%$

Profit percentage for B $= 20\%$

To Compare:

The profit earned by A and the profit earned by B.

Solution:

First, let's calculate the Selling Price of A (SP$_A$) and the profit earned by A.

A sells the tape recorder to B at a $20\%$ profit on A's cost price.

Profit for A $= 20\%$ of CP$_A$

Profit for A $= \frac{20}{100} \times 8000$

Profit for A $= 0.20 \times 8000$

Profit for A $= \textsf{₹} 1600$

The Selling Price of A (SP$_A$) is CP$_A$ + Profit for A.

$ \text{SP}_A = \textsf{₹} 8000 + \textsf{₹} 1600 $

$ \text{SP}_A = \textsf{₹} 9600 $

Now, B buys the tape recorder from A, so B's cost price (CP$_B$) is equal to A's selling price (SP$_A$).

CP$_B$ $= \textsf{₹} 9600$

B sells the tape recorder to C at a $20\%$ profit on B's cost price.

Profit for B $= 20\%$ of CP$_B$

Profit for B $= 20\%$ of $\textsf{₹} 9600$

Profit for B $= \frac{20}{100} \times 9600$

Profit for B $= 0.20 \times 9600$

Profit for B $= \textsf{₹} 1920$

Compare the profit earned by A and B:

Profit for A $= \textsf{₹} 1600$

Profit for B $= \textsf{₹} 1920$

Since $\textsf{₹} 1920 > \textsf{₹} 1600$, B earns more profit than A.

Equivalently, A earns less profit than B.

The correct statement is that A earns less profit than B.

The correct option is (c) A earns less profit than B.

Given:

Cost Price (CP) of teapot $= \textsf{₹} 120$

Cost Price (CP) of set of cups $= \textsf{₹} 400$

Teapot sold at a Profit of $5\%$

Cups sold at a Loss of $5\%$

To Find:

The total amount received by her (Total Selling Price).

Solution:

First, calculate the Selling Price (SP) of the teapot.

The teapot is sold at a profit of $5\%$.

Profit on teapot $= 5\% \text{ of CP of teapot}

Profit on teapot $= 5\% \text{ of } \textsf{₹} 120$

Profit on teapot $= \frac{5}{100} \times 120$

Profit on teapot $= 0.05 \times 120$

Profit on teapot $= \textsf{₹} 6$

Selling Price of teapot $= \text{CP of teapot} + \text{Profit on teapot}

Selling Price of teapot $= \textsf{₹} 120 + \textsf{₹} 6$

Selling Price of teapot $= \textsf{₹} 126$

Next, calculate the Selling Price (SP) of the set of cups.

The cups are sold at a loss of $5\%$.

Loss on cups $= 5\% \text{ of CP of cups}

Loss on cups $= 5\% \text{ of } \textsf{₹} 400$

Loss on cups $= \frac{5}{100} \times 400$

Loss on cups $= 0.05 \times 400$

Loss on cups $= \textsf{₹} 20$

Selling Price of cups $= \text{CP of cups} - \text{Loss on cups}

Selling Price of cups $= \textsf{₹} 400 - \textsf{₹} 20$

Selling Price of cups $= \textsf{₹} 380$

The total amount received by Latika is the sum of the selling prices of the teapot and the cups.

Total Amount Received $= \text{SP of teapot} + \text{SP of cups}

Total Amount Received $= \textsf{₹} 126 + \textsf{₹} 380$

Total Amount Received $= \textsf{₹} 506$

The total amount received by her is $\textsf{₹} 506$.

The correct option is (c) Rs 506.

Given:

Selling Price (SP) $= \textsf{₹} 1,120$

Discount Rate $= 20\%$

To Find:

The Marked Price (MP) of the jacket.

Solution:

When a discount of $20\%$ is allowed on the Marked Price, the Selling Price is the remaining percentage of the Marked Price.

Percentage paid $= 100\% - \text{Discount Rate} = 100\% - 20\% = 80\%$

So, the Selling Price is $80\%$ of the Marked Price.

$ \text{SP} = 80\% \text{ of MP} $

$ \textsf{₹} 1120 = \frac{80}{100} \times \text{MP} $

$ 1120 = 0.80 \times \text{MP} $

$ 1120 = 0.8 \times \text{MP} $

To find the Marked Price (MP), divide the Selling Price by $0.8$:

$ \text{MP} = \frac{1120}{0.8} $

To remove the decimal, multiply the numerator and denominator by 10:

$ \text{MP} = \frac{1120 \times 10}{0.8 \times 10} = \frac{11200}{8} $

Now, perform the division:

$ 11200 \div 8 $

$ 11 \div 8 = 1 $ remainder $3$

$ 32 \div 8 = 4 $ remainder $0$

$ 0 \div 8 = 0 $

$ 0 \div 8 = 0 $

So, $\frac{11200}{8} = 1400$.

$ \text{MP} = \textsf{₹} 1400 $

The marked price of the jacket is $\textsf{₹} 1400$.

The correct option is (b) Rs 1400.

Given:

Time period (n) $= 2$ years

Compounding Frequency: Every three months (Quarterly)

To Find:

The number of times interest is compounded in 2 years.

Solution:

Interest is compounded every three months. This means the interest is calculated and added to the principal at the end of every three-month period.

The number of months in a year is 12.

The duration of one compounding period is 3 months.

Number of compounding periods in one year $= \frac{\text{Number of months in a year}}{\text{Duration of one compounding period}}$

Number of compounding periods in one year $= \frac{12 \text{ months}}{3 \text{ months}} = 4$ periods.

Since the interest is compounded quarterly (every three months), there are 4 compounding periods in one year.

The total time period is 2 years.

Number of times interest is charged in 2 years = Number of compounding periods per year $\times$ Number of years

Total number of compounding periods $= 4 \text{ periods/year} \times 2 \text{ years} = 8$ periods.

The interest is charged 8 times in 2 years.

The correct option is (a) 8.

Given:

Total price of the washing machine including VAT $= \textsf{₹} 13,500$

VAT (Value Added Tax) Rate $= 8\%$

To Find:

The original price of the washing machine (price before VAT).

Solution:

Let the original price of the washing machine be $x$.

The VAT is calculated on the original price $x$.

VAT Amount $= 8\% \text{ of } x$

VAT Amount $= \frac{8}{100} \times x = 0.08x$

The total price paid is the sum of the original price and the VAT amount.

Total Price $= \text{Original Price} + \text{VAT Amount}

We are given that the Total Price is $\textsf{₹} 13,500$.

$ 13500 = x + 0.08x $

Combine the terms involving $x$:

$ 13500 = (1 + 0.08)x $

$ 13500 = 1.08x $

To find $x$ (the original price), divide both sides of the equation by $1.08$:

$ x = \frac{13500}{1.08} $

To remove the decimal from the denominator, multiply the numerator and denominator by 100:

$ x = \frac{13500 \times 100}{1.08 \times 100} = \frac{1350000}{108} $

Now, perform the division $\frac{1350000}{108}$. We can simplify by dividing both numbers by common factors.

Divide by 4:

$ 108 \div 4 = 27 $

$ 1350000 \div 4 = 337500 $

$ x = \frac{337500}{27} $

Now divide by 9 (since $3+3+7+5+0+0 = 18$, which is divisible by 9; and $2+7=9$, which is divisible by 9):

$ 27 \div 9 = 3 $

$ 337500 \div 9 $:

$ 33 \div 9 = 3 $ remainder $6$

$ 67 \div 9 = 7 $ remainder $4$

$ 45 \div 9 = 5 $ remainder $0$

$ 0 \div 9 = 0 $

$ 0 \div 9 = 0 $

So, $337500 \div 9 = 37500$.

$ x = \frac{37500}{3} $

Finally, divide by 3:

$ \frac{37500}{3} = 12500 $

$ x = 12500 $

The original price of the washing machine is $\textsf{₹} 12,500$.

The correct option is (c) Rs 12,500.

Given:

Iron 1: CP$_1 = \textsf{₹} 900$, Gain $\% = 10\%$

Iron 2: CP$_2 = \textsf{₹} 1200$, Loss $\% = 5\%$

To Find:

The net profit or loss on the entire transaction.

Solution:

First, calculate the Selling Price (SP$_1$) and Profit for Iron 1.

Profit on Iron 1 $= 10\%$ of CP$_1$

Profit on Iron 1 $= 10\%$ of $\textsf{₹} 900$

Profit on Iron 1 $= \frac{10}{100} \times 900 = 0.10 \times 900 = \textsf{₹} 90$

SP$_1 = \text{CP}_1 + \text{Profit}$

SP$_1 = \textsf{₹} 900 + \textsf{₹} 90 = \textsf{₹} 990$

Next, calculate the Selling Price (SP$_2$) and Loss for Iron 2.

Loss on Iron 2 $= 5\%$ of CP$_2$

Loss on Iron 2 $= 5\%$ of $\textsf{₹} 1200$

Loss on Iron 2 $= \frac{5}{100} \times 1200 = 0.05 \times 1200 = \textsf{₹} 60$

SP$_2 = \text{CP}_2 - \text{Loss}$

SP$_2 = \textsf{₹} 1200 - \textsf{₹} 60 = \textsf{₹} 1140$

Now, calculate the Total Cost Price (Total CP) and Total Selling Price (Total SP) for the entire transaction.

Total CP $= \text{CP}_1 + \text{CP}_2$

Total CP $= \textsf{₹} 900 + \textsf{₹} 1200 = \textsf{₹} 2100$

Total SP $= \text{SP}_1 + \text{SP}_2$

Total SP $= \textsf{₹} 990 + \textsf{₹} 1140 = \textsf{₹} 2130$

Compare the Total SP and Total CP:

Total SP $= \textsf{₹} 2130$

Total CP $= \textsf{₹} 2100$

Since Total SP $>$ Total CP, there is a net profit on the transaction.

Net Profit $= \text{Total SP} - \text{Total CP}$

Net Profit $= \textsf{₹} 2130 - \textsf{₹} 2100$

Net Profit $= \textsf{₹} 30$

Alternatively, we can find the net profit by summing the individual profit/loss amounts.

Net Change $= \text{Profit on Iron 1} + \text{Loss on Iron 2 (as a negative value)}

Net Change $= \textsf{₹} 90 + (-\textsf{₹} 60)

Net Change $= \textsf{₹} 90 - \textsf{₹} 60 = \textsf{₹} 30$

Since the net change is positive, it is a profit of $\textsf{₹} 30$.

On the transaction, he has a profit of $\textsf{₹} 30$.

The correct option is (c) Profit of Rs 30.

Given:

Total price of the TV set including VAT $= \textsf{₹} 26,250$

VAT (Value Added Tax) Rate $= 5\%$

To Find:

The original price of the TV set (price before VAT).

Solution:

Let the original price of the TV set be $x$.

The VAT is calculated on the original price $x$.

VAT Amount $= 5\% \text{ of } x$

VAT Amount $= \frac{5}{100} \times x = 0.05x$

The total price paid is the sum of the original price and the VAT amount.

Total Price $= \text{Original Price} + \text{VAT Amount}

We are given that the Total Price is $\textsf{₹} 26,250$.

$ 26250 = x + 0.05x $

Combine the terms involving $x$:

$ 26250 = (1 + 0.05)x $

$ 26250 = 1.05x $

To find $x$ (the original price), divide both sides of the equation by $1.05$:

$ x = \frac{26250}{1.05} $

To remove the decimal from the denominator, multiply the numerator and denominator by 100:

$ x = \frac{26250 \times 100}{1.05 \times 100} = \frac{2625000}{105} $

Now, perform the division $\frac{2625000}{105}$. We can simplify by dividing by common factors.

Divide by 5:

$ 105 \div 5 = 21 $

$ 2625000 \div 5 = 525000 $

$ x = \frac{525000}{21} $

Now, divide by 21. We can notice that $21 \times 25 = 525$.

$ 525 \div 21 = 25 $

So, $525000 \div 21 = 25000$.

$ x = 25000 $

The original price of the TV set is $\textsf{₹} 25,000$.

The correct option is (b) Rs 25,000.

Given Expression:

$ 40\% \text{ of } [100 – 20\% \text{ of } 300] $

To Evaluate:

The value of the given expression.

Solution:

We need to evaluate the expression step by step, following the order of operations (BODMAS/PEMDAS).

First, evaluate the term inside the square brackets: $[100 – 20\% \text{ of } 300]$.

Inside the bracket, we need to calculate $20\% \text{ of } 300$ first.

$ 20\% \text{ of } 300 = \frac{20}{100} \times 300 $

$ 20\% \text{ of } 300 = 0.20 \times 300 $

$ 20\% \text{ of } 300 = \textsf{₹} 60 $

Now substitute this value back into the expression inside the brackets:

$ [100 – 20\% \text{ of } 300] = [100 - 60] = 40 $

Now the original expression becomes:

$ 40\% \text{ of } 40 $

Calculate $40\%$ of $40$:

$ 40\% \text{ of } 40 = \frac{40}{100} \times 40 $

$ 40\% \text{ of } 40 = 0.40 \times 40 $

$ 40\% \text{ of } 40 = 16 $

The value of the expression is $16$.

The correct option is (b) 16.

Given:

Original Price of the car (P$_0$) $= \textsf{₹} 2,50,000$

Decrease in price in the first year $= 10\%$

Decrease in price in the second year $= 12\%$ (This decrease is on the price after the first year)

To Find:

The overall decrease per cent in the price of the car over two years.

Solution:

First, calculate the price of the car after the first year.

Decrease in the first year $= 10\%$ of P$_0$

Decrease in Year 1 $= 10\%$ of $\textsf{₹} 2,50,000$

Decrease in Year 1 $= \frac{10}{100} \times 250000 $

Decrease in Year 1 $= 0.10 \times 250000 $

Decrease in Year 1 $= \textsf{₹} 25000$

Price after Year 1 (P$_1$) $= \text{Original Price} - \text{Decrease in Year 1}$

P$_1 = \textsf{₹} 250000 - \textsf{₹} 25000$

P$_1 = \textsf{₹} 225000$

Next, calculate the price of the car after the second year. The decrease in the second year is $12\%$ of the price at the end of the first year (P$_1$).

Decrease in the second year $= 12\%$ of P$_1$

Decrease in Year 2 $= 12\%$ of $\textsf{₹} 2,25,000$

Decrease in Year 2 $= \frac{12}{100} \times 225000$

Decrease in Year 2 $= 0.12 \times 225000$

Decrease in Year 2 $= \textsf{₹} 27000$

Price after Year 2 (P$_2$) $= \text{Price after Year 1} - \text{Decrease in Year 2}$

P$_2 = \textsf{₹} 225000 - \textsf{₹} 27000$

P$_2 = \textsf{₹} 198000$

Now, calculate the total decrease in price over the two years.

Total Decrease $= \text{Original Price} - \text{Price after 2 years}$

Total Decrease $= \textsf{₹} 250000 - \textsf{₹} 198000$

Total Decrease $= \textsf{₹} 52000$

Finally, calculate the overall decrease per cent. This is the total decrease expressed as a percentage of the original price.

Overall Decrease $\% = \frac{\text{Total Decrease}}{\text{Original Price}} \times 100\%$

Overall Decrease $\% = \frac{\textsf{₹} 52000}{\textsf{₹} 250000} \times 100\%$

Overall Decrease $\% = \frac{52000}{250000} \times 100\%$

Cancel common zeros:

Overall Decrease $\% = \frac{52}{250} \times 100\%$

Simplify the fraction:

Overall Decrease $\% = \frac{\cancel{52}^{26}}{\cancel{250}_{125}} \times 100\%$

Overall Decrease $\% = \frac{26}{125} \times 100\%$

Simplify further:

Overall Decrease $\% = \frac{26}{\cancel{125}_{5}} \times \cancel{100}^{4}\%$

Overall Decrease $\% = \frac{26 \times 4}{5}\%$

Overall Decrease $\% = \frac{104}{5}\%$

Perform the division $104 \div 5$:

$ 104 \div 5 = 20.8 $

Overall Decrease $\% = 20.8\%$

Alternate Method (Using Successive Percentage Change):

If a quantity is first decreased by $d_1\%$ and then decreased by $d_2\%$, the net percentage change is given by the formula:

$ \text{Net Change} \% = -d_1 - d_2 + \frac{d_1 \times d_2}{100} $

Here, $d_1 = 10$ and $d_2 = 12$. Both are decreases, so we can use the formula for successive changes as decreases.

Effective decrease $\%= d_1 + d_2 - \frac{d_1 \times d_2}{100}$

Effective decrease $\%= 10 + 12 - \frac{10 \times 12}{100}$

Effective decrease $\%= 22 - \frac{120}{100}$

Effective decrease $\%= 22 - 1.2$

Effective decrease $\%= 20.8$

The overall decrease per cent is $20.8\%$.

The overall decrease per cent in the price of the car is $20.8\%$.

The correct option is (c) 20.8%.

The blank should be filled with the word "Discount".

Discount is indeed a reduction given on the marked price (also known as list price or tag price) of an article.

The marked price is the price printed on the tag of the article.

The selling price is the actual price at which the article is sold to the customer.

The relationship between Marked Price (MP), Selling Price (SP), and Discount is:

Discount = Marked Price - Selling Price

Often, discount is calculated as a percentage of the marked price. If the discount percentage is $D\%$, then the discount amount is:

Discount Amount $= \frac{D}{100} \times \text{Marked Price}$

And the Selling Price would be:

Selling Price = Marked Price - Discount Amount

Selling Price $= \text{Marked Price} - \frac{D}{100} \times \text{Marked Price}$

Selling Price $= \text{Marked Price} \times \left(1 - \frac{D}{100}\right)$

In summary, the term that represents a reduction from the original price (marked price) of an item to arrive at the final selling price is Discount.

The blank should be filled with the number 8.

Given:

Original Value = 150

New Value = 162

To Find:

Percentage Increase.

Solution:

First, calculate the amount of increase:

Amount of Increase = New Value - Original Value

Amount of Increase $= 162 - 150$

Amount of Increase $= 12$

Next, calculate the percentage increase using the formula:

Percentage Increase $= \left(\frac{\text{Amount of Increase}}{\text{Original Value}}\right) \times 100\%$

Percentage Increase $= \left(\frac{12}{150}\right) \times 100\%$

Percentage Increase $= \frac{12}{150} \times 100$

We can simplify the expression:

$\frac{12}{150} \times 100 = \frac{\cancel{12}^{2}}{\cancel{150}_{25}} \times 100$ (Dividing 12 and 150 by their HCF, which is 6)

$= \frac{2}{25} \times 100$

$= \frac{2}{\cancel{25}_1} \times \cancel{100}^{4}$ (Dividing 25 and 100 by 25)

$= 2 \times 4$

$= 8$

Alternatively,

Percentage Increase $= \frac{12}{150} \times 100$

$= \frac{12 \times 100}{150}$

$= \frac{1200}{150}$

$= \frac{\cancel{1200}^{120}}{\cancel{150}_{15}}$ (Dividing numerator and denominator by 10)

$= \frac{\cancel{120}^{8}}{\cancel{15}_{1}}$ (Dividing numerator and denominator by 15)

$= 8$

Therefore, the percentage increase is $8\%$.

The increase of a number from 150 to 162 is equal to increase of 8 per cent.

The blank should be filled with the amount 243.

Given:

Original Price of the article = $\textsf{₹}$ 1,620

Percentage Increase = 15%

To Find:

The amount of increase in Rupees.

Solution:

The increase in price is 15% of the original price.

Amount of Increase $= 15\%$ of $\textsf{₹}$ 1,620

Amount of Increase $= \frac{15}{100} \times 1620$

Amount of Increase $= \frac{15 \times 1620}{100}$

Amount of Increase $= \frac{24300}{100}$

Amount of Increase $= 243$

Alternatively, simplify the fraction before multiplying:

Amount of Increase $= \frac{15}{100} \times 1620$

Cancel out a 10 from the numerator and denominator:

Amount of Increase $= \frac{15}{\cancel{100}_{10}} \times \cancel{1620}^{162}$

$= \frac{15}{10} \times 162$

$= \frac{\cancel{15}^{3}}{\cancel{10}_{2}} \times 162$ (Dividing 15 and 10 by 5)

$= \frac{3}{2} \times 162$

$= 3 \times \frac{162}{2}$

$= 3 \times \cancel{81}$ (Dividing 162 by 2)

$= 243$

The amount of increase is $\textsf{₹}$ 243.

15% increase in price of an article, which is $\textsf{₹}$ 1,620, is the increase of $\textsf{₹}$ 243.

The blanks should be filled with Marked Price and Selling Price respectively.

Discount is the reduction in price offered on the Marked Price of an article.

The price at which the article is finally sold after the discount is called the Selling Price.

The relationship between these three quantities is given by the formula:

Discount = Marked Price - Selling Price

So, the completed statement is:

Discount = Marked Price – Selling Price.

The blank should be filled with Marked Price.

Discount is the reduction offered on the original price of an article.

The price mentioned on the label or tag of an article is called the Marked Price.

When a discount is offered, it is usually given as a percentage of this Marked Price.

If the Discount Percentage is $D\%$, the actual amount of discount is calculated on the Marked Price (MP).

The formula for calculating the discount amount is:

Discount Amount $= \frac{\text{Discount Percentage}}{100} \times \text{Marked Price}$

This can be written as:

Discount $= \text{Discount} \% \text{ of Marked Price}$

Therefore, the statement should be:

Discount = Discount % of Marked Price.

The blank should be filled with Sales Tax or Value Added Tax (VAT) or Goods and Services Tax (GST) depending on the specific taxation system in place, but commonly referred to as Sales Tax in a general context.

When you buy goods or services, the government often levies a tax on the sale. This tax is calculated as a percentage of the selling price of the item.

This amount of tax is added to the selling price to arrive at the final price that the customer has to pay.

For example, if an item has a selling price of $\textsf{₹} 100$ and the sales tax is $10\%$, the amount of sales tax would be $10\%$ of $\textsf{₹} 100$, which is $\textsf{₹} 10$.

The final bill amount would then be $\textsf{₹} 100 + \textsf{₹} 10 = \textsf{₹} 110$.

Different terms like Sales Tax, Value Added Tax (VAT), or Goods and Services Tax (GST) are used in various countries and regions to describe this type of tax. However, the underlying principle is the same: it is a tax on consumption or sales, collected by the seller and paid to the government, which increases the final price for the consumer.

In the context of a simple fill-in-the-blank question, Sales Tax is a commonly accepted answer.

So, the completed statement is:

Sales Tax is charged on the sale of an item by the government and is added to the bill amount.

The blank should be filled with the formula for the amount under annual compounding.

The formula for the Amount ($A$) when the principal ($P$) is compounded annually at an interest rate of $R\%$ per annum for $n$ years is:

$A = P \left(1 + \frac{R}{100}\right)^n$

In this formula:

$A$ stands for the Amount (Principal + Compound Interest)

$P$ stands for the Principal amount (the initial sum of money)

$R$ stands for the Rate of Interest per annum (usually expressed as a percentage)

$n$ stands for the Number of Years (the time period for which the interest is compounded)

So, the completed statement is:

Amount when interest is compounded annually is given by the formula $\mathbf{A = P (1 + \frac{R}{100})^n}$.

The blank should be filled with Selling Price or Value.

Sales tax is a tax levied by the government on the sale of goods and services.

This tax is typically calculated as a percentage of the price at which the item is sold to the customer, which is the Selling Price.

If the tax percentage is $T\%$, the amount of sales tax is calculated using the formula:

Sales Tax Amount $= T\%$ of Selling Price

Sales Tax Amount $= \frac{T}{100} \times \text{Selling Price}$

This sales tax amount is then added to the selling price to get the final amount the customer pays.

So, the completed statement is:

Sales tax = tax % of Selling Price.

The blank should be filled with compounding period or conversion period.

In compound interest calculations, the interest earned is not paid out but is added to the principal amount at the end of specific intervals. This new amount then becomes the principal for the next interest calculation period.

The duration of this specific interval, after which the interest is added to the principal, is called the compounding period or the conversion period.

Common compounding periods include:

- Annually (once a year)

- Half-yearly or Semi-annually (twice a year)

- Quarterly (four times a year)

- Monthly (twelve times a year)

The frequency of compounding directly affects the total amount of interest earned, as interest is calculated on a growing principal balance.

So, the completed statement is:

The time period after which the interest is added each time to form a new principal is called the compounding period.

The blank should be filled with Overhead or Additional.

When a buyer purchases an item, the initial cost is the price paid to the seller (the cost of purchase or cost price).

However, there can be other expenses incurred to make the item ready for use or sale, such as transportation costs, labour charges for installation, repair costs, etc.

These additional expenses are added to the cost of purchase to determine the total cost or total cost price of the item for the buyer.

These additional expenses are commonly known as Overhead Expenses or simply Additional Expenses.

Total Cost Price = Cost of Purchase + Overhead Expenses

So, the completed statement is:

Overhead expenses are the additional expenses incurred by a buyer for an item over and above its cost of purchase.

The blank should be filled with Marked Price.

The discount offered on an item is always calculated based on the original price listed for the item. This original listed price is known as the Marked Price or List Price.

It is the price that is usually displayed on the product tag or label before any discount is applied.

If a shopkeeper announces a discount of, say, 10%, it means the reduction in price will be 10% of the Marked Price.

The formula for discount amount is:

Discount Amount $= \frac{\text{Discount Percentage}}{100} \times \text{Marked Price}$

The final selling price is then calculated by subtracting the discount amount from the marked price:

Selling Price = Marked Price - Discount Amount

Therefore, the basis for calculating the discount is the Marked Price.

So, the completed statement is:

The discount on an item for sale is calculated on the Marked Price.

The blank should be filled with the formula for the amount under semi-annual compounding.

When interest is compounded semi-annually:

The annual interest rate ($r\%$) is divided by 2 to get the rate per compounding period ($\frac{r}{2}\%$).

The number of years ($t$) is multiplied by 2 to get the total number of compounding periods ($2t$).

Using the general compound interest formula $A = P \left(1 + \frac{R}{100}\right)^n$, where $R$ is the rate per period and $n$ is the number of periods:

Substitute $R = \frac{r}{2}$ and $n = 2t$.

Amount ($A$) $= P \left(1 + \frac{r/2}{100}\right)^{2t}$

Amount ($A$) $= P \left(1 + \frac{r}{2 \times 100}\right)^{2t}$

Amount ($A$) $= P \left(1 + \frac{r}{200}\right)^{2t}$

In this formula:

$P$ is the Principal.

$r$ is the annual rate of interest.

$t$ is the time period in years.

$A$ is the Amount after $t$ years.

So, the completed statement is:

When principal P is compounded semi-annually at r % per annum for t years, then Amount = $\mathbf{P \left(1 + \frac{r}{200}\right)^{2t}}$.

The blanks should be filled with related (or equivalent) and denominator.

A percentage is a way of expressing a number as a fraction of 100.

The term "percent" comes from the Latin "per centum", meaning "out of one hundred".

For example:

$25\%$ is equivalent to the fraction $\frac{25}{100}$.

$50\%$ is equivalent to the fraction $\frac{50}{100}$.

$7\%$ is equivalent to the fraction $\frac{7}{100}$.

In each case, the percentage is directly related or equivalent to a fraction where the numerator is the percentage value and the denominator is 100.

Therefore, the statement means that percentages can be expressed as fractions where the lower part (the denominator) is always 100.

So, the completed statement is:

Percentages are related to fractions with denominator equal to 100.

The blank should be filled with the amount 1000.

Given:

Selling Price (SP) $= \textsf{₹}$ 880

Discount Percentage (D%) = 12%

To Find:

Marked Price (MP).

Solution:

The Selling Price is obtained after giving a discount on the Marked Price.

The discount amount is calculated as a percentage of the Marked Price.

Discount Amount $= 12\%$ of Marked Price

Discount Amount $= \frac{12}{100} \times \text{MP}$

The relationship between Selling Price, Marked Price, and Discount is:

Selling Price = Marked Price - Discount Amount

Substituting the expression for Discount Amount:

SP = MP $- \left(\frac{12}{100} \times \text{MP}\right)$

SP = MP $\left(1 - \frac{12}{100}\right)$

SP = MP $\left(\frac{100 - 12}{100}\right)$

SP = MP $\left(\frac{88}{100}\right)$

Now, we need to find MP. Rearranging the formula:

MP $= SP \times \frac{100}{88}$

Substitute the given value of SP = 880:

MP $= 880 \times \frac{100}{88}$

We can simplify this calculation:

MP $= \frac{\cancel{880}^{10} \times 100}{\cancel{88}_{1}}$ (Dividing 880 by 88 gives 10)

MP $= 10 \times 100$

MP $= 1000$

Thus, the marked price of the article is $\textsf{₹}$ 1000.

The marked price of an article when it is sold for $\textsf{₹}$ 880 after a discount of 12% is $\textsf{₹}$ 1000.

The blank should be filled with the amount 1331.20.

Given:

Principal (P) = $\textsf{₹}$ 8,000

Time (t) = 1 year

Annual Rate of Interest (r) = 16% p.a.

Compounding Frequency = Half-yearly

Given value: $(1.08)^2 = 1.1664$

To Find:

Compound Interest (CI).

Solution:

When interest is compounded half-yearly, we need to adjust the rate and time.

The rate per half-year period ($R$) is half of the annual rate:

$R = \frac{\text{Annual Rate}}{2} = \frac{16\%}{2} = 8\%$

In decimal form, $R = \frac{8}{100} = 0.08$.

The number of compounding periods ($n$) in the given time is the number of half-years in 1 year:

$n = \text{Number of years} \times 2 = 1 \times 2 = 2$ periods

The formula for the Amount (A) when compounded is $A = P (1 + R)^n$, where $R$ is the rate per period and $n$ is the number of periods.

$A = 8000 \left(1 + 0.08\right)^2$

$A = 8000 (1.08)^2$

Using the given value $(1.08)^2 = 1.1664$:

$A = 8000 \times 1.1664$

$A = 9331.20$

This is the Amount (Principal + Compound Interest). To find the Compound Interest (CI), we subtract the Principal from the Amount:

CI = Amount - Principal

CI = $\textsf{₹} 9331.20 - \textsf{₹} 8000$

CI = $\textsf{₹} 1331.20$

The compound interest is $\textsf{₹}$ 1331.20.

The compound interest on $\textsf{₹}$ 8,000 for one year at 16% p.a. compounded half yearly is $\textsf{₹}$ 1331.20, given that $(1.08)^2 = 1.1664$.

The blank should be filled with gain of $\textsf{₹}$ 27,000.

Given:

Initial Investment (Principal) = $\textsf{₹}$ 6,00,000

Result in Year 1 = 5% Loss

Result in Year 2 = 10% Gain

To Find:

The net result (overall gain or loss).

Solution:

First, calculate the value of the investment after the first year's loss.

Loss in Year 1 = 5% of $\textsf{₹}$ 6,00,000

Loss in Year 1 $= \frac{5}{100} \times 600000$

Loss in Year 1 $= 5 \times 6000$

Loss in Year 1 $= \textsf{₹}$ 30,000

Value at the end of Year 1 = Initial Investment - Loss in Year 1

Value at the end of Year 1 $= 600000 - 30000$

Value at the end of Year 1 $= \textsf{₹}$ 5,70,000

Next, calculate the value of the investment after the second year's gain. The gain in the second year is calculated on the value at the beginning of the second year, which is the value at the end of Year 1.

Gain in Year 2 = 10% of $\textsf{₹}$ 5,70,000

Gain in Year 2 $= \frac{10}{100} \times 570000$

Gain in Year 2 $= \frac{1}{10} \times 570000$

Gain in Year 2 $= \textsf{₹}$ 57,000

Value at the end of Year 2 = Value at the end of Year 1 + Gain in Year 2

Value at the end of Year 2 $= 570000 + 57000$

Value at the end of Year 2 $= \textsf{₹}$ 6,27,000

Finally, calculate the net result by comparing the final value with the initial investment.

Net Change = Final Value - Initial Investment

Net Change $= 627000 - 600000$

Net Change $= \textsf{₹}$ 27,000

Since the net change is positive, it represents a gain.

Net Result = Gain of $\textsf{₹}$ 27,000.

The net result is a gain of $\textsf{₹}$ 27,000.

The first blank should be filled with 10% and the second blank with 1.5 (or $1\frac{1}{2}$).

Given:

Principal (P) = $\textsf{₹}$ 6,000

Amount formula structure: $6000 \left( 1+\frac{5}{100} \right)^{3}$

Compounding frequency: Half yearly

To Find:

Rate of interest per annum (p.a.) and time in years.

Solution:

The general formula for the Amount ($A$) when interest is compounded $k$ times a year is:

$A = P \left(1 + \frac{r/k}{100}\right)^{kt}$

where $P$ is the principal, $r$ is the annual rate of interest (in percentage), $t$ is the time in years, and $k$ is the number of times interest is compounded per year.

In this problem, the compounding is half yearly, which means $k=2$ (twice a year).

The given formula structure is $6000 \left( 1+\frac{5}{100} \right)^{3}$.

Comparing this with the general formula for half-yearly compounding ($k=2$):

$A = P \left(1 + \frac{r/2}{100}\right)^{2t}$

The given expression is $A = 6000 \left( 1+\frac{5}{100} \right)^{3}$.

By comparing the terms inside the parenthesis:

$\frac{r/2}{100} = \frac{5}{100}$

This implies $\frac{r}{2} = 5$.

Solving for $r$:

$r = 5 \times 2$

$r = 10$

So, the annual rate of interest is 10%.

By comparing the exponents:

$2t = 3$

Solving for $t$:

$t = \frac{3}{2}$

$t = 1.5$

So, the time in years is 1.5 years.

The rate of interest p.a. is 10% and the time in years is 1.5.

If amount on the principal of $\textsf{₹}$ 6,000 is written as $6000 \left( 1+\frac{5}{100} \right)^{3}$ and compound interest payable half yearly, then rate of interest p.a. is 10% and time in years is 1.5.

The blank should be filled with the amount 80,000.

Given:

Selling Price (SP) = $\textsf{₹}$ 1,12,000

Gain Percentage (Gain%) = 40%

To Find:

Cost Price (CP).

Solution:

When an article is sold at a gain of 40%, it means the selling price is 40% more than the cost price.

Selling Price = Cost Price + Gain

Gain = 40% of Cost Price

Gain $= \frac{40}{100} \times \text{CP} = 0.40 \times \text{CP}$

Substituting the gain into the selling price equation:

SP = CP + $0.40 \times$ CP

SP = CP $(1 + 0.40)$

SP = CP $(1.40)$

We are given SP = $\textsf{₹}$ 1,12,000. Substitute this value into the equation:

$112000 = \text{CP} \times 1.40$

Now, solve for CP:

CP $= \frac{112000}{1.40}$

To perform the division, remove the decimal by multiplying the numerator and denominator by 100:

CP $= \frac{112000 \times 100}{1.40 \times 100}$

CP $= \frac{11200000}{140}$

Cancel out a zero from numerator and denominator:

CP $= \frac{1120000}{\cancel{140}_{14}}$

CP $= \frac{1120000}{14}$

Divide 112 by 14 ($14 \times 8 = 112$):

CP $= \frac{\cancel{112}^{8}0000}{\cancel{14}_{1}}$

CP $= 80000$

Alternatively, if the selling price is $100\% + 40\% = 140\%$ of the cost price:

$140\%$ of CP $= \textsf{₹}$ 1,12,000

$\frac{140}{100} \times \text{CP} = 112000$

CP $= 112000 \times \frac{100}{140}$

CP $= 112000 \times \frac{10}{14}$

CP $= \frac{\cancel{112000}^{8000} \times 10}{\cancel{14}_{1}}$ (Dividing 112000 by 14)

CP $= 8000 \times 10$

CP $= 80000$

The cost price of the article was $\textsf{₹}$ 80,000.

By selling an article for $\textsf{₹}$ 1,12,000 a girl gains 40%. The cost price of the article was $\textsf{₹}$ 80,000.

The blank should be filled with the percentage $\mathbf{6\frac{2}{3}\%}$ or $\mathbf{\frac{20}{3}\%}$.

Given:

Number of geometry boxes sold = 140

Loss = Selling Price (S.P.) of 10 geometry boxes.

To Find:

Loss Percentage.

Solution:

Let SP be the Selling Price of one geometry box.

Let CP be the Cost Price of one geometry box.

The Selling Price of 140 geometry boxes is $140 \times \text{SP}$.

The Cost Price of 140 geometry boxes is $140 \times \text{CP}$.

The given Loss is equal to the Selling Price of 10 geometry boxes, which is $10 \times \text{SP}$.

The relationship between Cost Price, Selling Price, and Loss is:

Loss = Cost Price - Selling Price

For 140 geometry boxes:

Loss on 140 boxes = CP of 140 boxes - SP of 140 boxes

Substitute the given loss and the total CP and SP:

$10 \times \text{SP} = 140 \times \text{CP} - 140 \times \text{SP}$

Rearrange the equation to find the relationship between CP and SP:

$10 \times \text{SP} + 140 \times \text{SP} = 140 \times \text{CP}$

$150 \times \text{SP} = 140 \times \text{CP}$

The Loss Percentage is calculated on the Cost Price (CP):

Loss \% $= \left(\frac{\text{Loss}}{\text{Cost Price}}\right) \times 100\%$

The Loss is $10 \times \text{SP}$.

The Cost Price of the 140 boxes being sold is $140 \times \text{CP}$.

Loss \% $= \left(\frac{10 \times \text{SP}}{140 \times \text{CP}}\right) \times 100\%$

From the relationship $150 \times \text{SP} = 140 \times \text{CP}$, we can substitute $140 \times \text{CP}$ with $150 \times \text{SP}$ in the denominator:

Loss \% $= \left(\frac{10 \times \text{SP}}{150 \times \text{SP}}\right) \times 100\%$

Cancel out SP (assuming SP is not zero):

Loss \% $= \left(\frac{10}{150}\right) \times 100\%$

Loss \% $= \frac{\cancel{10}^{1}}{\cancel{150}_{15}} \times 100\%$

Loss \% $= \frac{1}{15} \times 100\%$

Loss \% $= \frac{100}{15}\%$

Simplify the fraction:

Loss \% $= \frac{\cancel{100}^{20}}{\cancel{15}_{3}}\%$ (Dividing numerator and denominator by 5)

Loss \% $= \frac{20}{3}\%$

This can also be written as a mixed number:

$\frac{20}{3} = 6$ with a remainder of $2$.

Loss \% $= 6\frac{2}{3}\%$

The loss per cent is $6\frac{2}{3}\%$.

The loss per cent on selling 140 geometry boxes at the loss of S.P. of 10 geometry boxes is equal to $6\frac{2}{3}\%$.

The blank should be filled with the percentage 100%.

Given:

Cost Price of 10 tables = Selling Price of 5 tables.

To Find:

Profit Percentage.

Solution:

Let CP be the Cost Price of one table.

Let SP be the Selling Price of one table.

According to the given information:

Cost Price of 10 tables $= 10 \times \text{CP}$

Selling Price of 5 tables $= 5 \times \text{SP}$

We are given that these two are equal:

We want to find the profit percentage. Profit or loss is usually calculated based on the Cost Price of the items sold. Let's consider the transaction of selling 5 tables.

Cost Price of 5 tables $= 5 \times \text{CP}$

Selling Price of 5 tables $= 5 \times \text{SP}$

From the given relationship $10 \times \text{CP} = 5 \times \text{SP}$, we can see that the Selling Price of 5 tables is equal to the Cost Price of 10 tables.

So, SP of 5 tables $= 10 \times \text{CP}$.

Now we can calculate the profit made on selling 5 tables:

Profit on 5 tables = Selling Price of 5 tables - Cost Price of 5 tables

Profit on 5 tables $= (10 \times \text{CP}) - (5 \times \text{CP})$

Profit on 5 tables $= 5 \times \text{CP}$

The Profit Percentage is calculated on the Cost Price of the items sold (which is 5 tables in this context):

Profit \% $= \left(\frac{\text{Profit on 5 tables}}{\text{Cost Price of 5 tables}}\right) \times 100\%$

Profit \% $= \left(\frac{5 \times \text{CP}}{5 \times \text{CP}}\right) \times 100\%$

Profit \% $= \left(\frac{\cancel{5 \times \text{CP}}}{\cancel{5 \times \text{CP}}}\right) \times 100\%$

Profit \% $= 1 \times 100\%$

Profit \% $= 100\%$

Alternatively, from $10 \times \text{CP} = 5 \times \text{SP}$, we can find the relationship between SP and CP for a single table:

$\text{SP} = \frac{10}{5} \times \text{CP} = 2 \times \text{CP}$

The selling price of one table is double its cost price.

Profit per table = SP - CP $= 2 \times \text{CP} - \text{CP} = \text{CP}$

Profit \% $= \left(\frac{\text{Profit per table}}{\text{CP per table}}\right) \times 100\%$

Profit \% $= \left(\frac{\text{CP}}{\text{CP}}\right) \times 100\% = 100\%$

The profit per cent in this transaction is 100%.

The cost price of 10 tables is equal to the sale price of 5 tables. The profit per cent in this transaction is 100%.

The blank should be filled with the amount 364.

Given:

Number of pens bought = 100

Rate per pen = $\textsf{₹}$ 3.50

Sales Tax Rate = 4%

To Find:

Total amount paid by Abida.

Solution:

First, calculate the cost price of the 100 pens before adding tax.

Cost of 100 pens = Number of pens $\times$ Rate per pen

Cost of 100 pens $= 100 \times \textsf{₹} 3.50$

Cost of 100 pens $= \textsf{₹} 350$

Next, calculate the amount of sales tax. The sales tax is 4% of the cost price.

Sales Tax Amount $= 4\%$ of Cost of 100 pens

Sales Tax Amount $= \frac{4}{100} \times 350$

Sales Tax Amount $= \frac{4 \times 350}{100}$

Sales Tax Amount $= \frac{1400}{100}$

Sales Tax Amount $= 14$

The sales tax amount is $\textsf{₹}$ 14.

Finally, calculate the total amount paid by adding the sales tax to the cost price.

Total Amount Paid = Cost of 100 pens + Sales Tax Amount

Total Amount Paid $= \textsf{₹} 350 + \textsf{₹} 14$

Total Amount Paid $= \textsf{₹} 364$

The total amount paid by Abida is $\textsf{₹}$ 364.

Abida bought 100 pens at the rate of $\textsf{₹}$ 3.50 per pen and pays a sales tax of 4%. The total amount paid by Abida is $\textsf{₹}$ 364.

The blank should be filled with the amount 10,000.

Given:

Cost of tape-recorder inclusive of sales tax = $\textsf{₹}$ 10,800

Sales Tax Rate = 8%

To Find:

Price of the tape-recorder before sales tax was charged.

Solution:

Let the price of the tape-recorder before sales tax be $\textsf{₹} P$.

The sales tax is charged at 8% on the price before tax.

Sales Tax Amount $= 8\%$ of $P$

Sales Tax Amount $= \frac{8}{100} \times P = 0.08 P$

The cost of the tape-recorder inclusive of sales tax is the sum of the price before tax and the sales tax amount.

Cost (inclusive of tax) = Price before tax + Sales Tax Amount

$10800 = P + 0.08 P$

$10800 = P (1 + 0.08)$

$10800 = P (1.08)$

To find the price before tax ($P$), we need to divide the total cost by 1.08.

$P = \frac{10800}{1.08}$

To simplify the division, we can multiply the numerator and the denominator by 100 to remove the decimal from the denominator:

$P = \frac{10800 \times 100}{1.08 \times 100}$

$P = \frac{1080000}{108}$

Now, perform the division:

$1080000 \div 108 = 10000$

$P = 10000$

So, the price of the tape-recorder before sales tax was $\textsf{₹}$ 10,000.

We can verify this:

Sales tax on $\textsf{₹}$ 10,000 at 8% $= \frac{8}{100} \times 10000 = \textsf{₹} 800$.

Cost inclusive of tax $= \textsf{₹} 10000 + \textsf{₹} 800 = \textsf{₹} 10800$.

This matches the given cost.

The price of the tape-recorder before sales tax was charged is $\textsf{₹}$ 10,000.

The blank should be filled with the percentage 400.

Given:

The two numbers are 2500 and 500.

To Find:

By what percentage is 2500 greater than 500.

Solution:

We need to find the percentage increase from the number 500 to the number 2500. The original value is 500, and the amount of increase is the difference between 2500 and 500.

Amount of Increase = $2500 - 500$

Amount of Increase $= 2000$

The percentage increase is calculated relative to the original value (500).

Percentage Increase $= \left(\frac{\text{Amount of Increase}}{\text{Original Value}}\right) \times 100\%$

Percentage Increase $= \left(\frac{2000}{500}\right) \times 100\%$

Percentage Increase $= \frac{2000}{500} \times 100$

Simplify the fraction $\frac{2000}{500}$:

$\frac{\cancel{2000}^{4}}{\cancel{500}_{1}} = 4$ (Dividing numerator and denominator by 500)

Percentage Increase $= 4 \times 100\%$

Percentage Increase $= 400\%$

Thus, 2500 is greater than 500 by 400%.

2500 is greater than 500 by 400%.

The blank should be filled with the percentage 300.

Given:

A number is multiplied by four.

To Find:

The percentage increase from the original number to four times the number.

Solution:

Let the original number be $x$.

Four times the number is $4x$.

The increase in the number is the difference between the new value and the original value.

Increase $= \text{New Value} - \text{Original Value}$

Increase $= 4x - x$

Increase $= 3x$

The percentage increase is calculated relative to the original number ($x$).

Percentage Increase $= \left(\frac{\text{Increase}}{\text{Original Value}}\right) \times 100\%$

Percentage Increase $= \left(\frac{3x}{x}\right) \times 100\%$

Assuming the original number $x$ is not zero, we can cancel $x$ from the numerator and denominator.

Percentage Increase $= 3 \times 100\%$

Percentage Increase $= 300\%$

Thus, four times a number represents a 300% increase in the number.

For example, if the number is 100, four times the number is 400. The increase is $400 - 100 = 300$. The percentage increase is $\left(\frac{300}{100}\right) \times 100\% = 3 \times 100\% = 300\%$.

Four times a number is a 300 % increase in the number.

The blank should be filled with the amount 199.50.

Given:

Marked Price (MP) = $\textsf{₹}$ 200

Discount Percentage (D%) = 5%

Sales Tax Percentage (ST%) = 5%

To Find:

Total amount payable after discount and sales tax.

Solution:

First, calculate the amount of discount given on the Marked Price.

Discount Amount $= 5\%$ of Marked Price

Discount Amount $= \frac{5}{100} \times 200$

Discount Amount $= \frac{5 \times 200}{100}$

Discount Amount $= \frac{1000}{100}$

Discount Amount $= \textsf{₹} 10$

Next, calculate the Selling Price (SP) after deducting the discount from the Marked Price.

Selling Price = Marked Price - Discount Amount

Selling Price $= \textsf{₹} 200 - \textsf{₹} 10$

Selling Price $= \textsf{₹} 190$

Now, calculate the amount of sales tax charged on the Selling Price.

Sales Tax Amount $= 5\%$ of Selling Price

Sales Tax Amount $= \frac{5}{100} \times 190$

Sales Tax Amount $= \frac{5 \times 190}{100}$

Sales Tax Amount $= \frac{950}{100}$

Sales Tax Amount $= 9.50$

The sales tax amount is $\textsf{₹} 9.50$.

Finally, calculate the total amount payable by adding the sales tax amount to the Selling Price.

Total Amount Payable = Selling Price + Sales Tax Amount

Total Amount Payable $= \textsf{₹} 190 + \textsf{₹} 9.50$

Total Amount Payable $= \textsf{₹} 199.50$

The amount payable is $\textsf{₹} 199.50$.

5% sales tax is charged on an article marked $\textsf{₹}$ 200 after allowing a discount of 5%, then the amount payable is $\textsf{₹}$ 199.50.

The statement is True.

Compound interest involves the principle of exponential growth, where the interest earned in each period is added to the principal, and subsequent interest is calculated on the new, larger amount.

Bacterial growth often follows an exponential pattern, where the population increases by a certain percentage in a given time interval, and the growth in the next interval is based on the increased population.

The formula for the Amount ($A$) under compound interest is $A = P \left(1 + \frac{R}{100}\right)^n$, where $P$ is the initial principal, $R$ is the rate per period, and $n$ is the number of periods.

Similarly, the formula for exponential growth of a quantity (like bacteria) is $N_t = N_0 (1 + r)^t$, where $N_t$ is the quantity at time $t$, $N_0$ is the initial quantity, $r$ is the growth rate per unit time, and $t$ is the time periods.

These formulas are mathematically equivalent. The initial number of bacteria ($N_0$) is analogous to the principal ($P$), the growth rate per period ($r$) is analogous to the interest rate per period ($\frac{R}{100}$), and the number of time periods ($t$) is analogous to the number of compounding periods ($n$).

Therefore, the formula for compound interest can indeed be used to calculate the growth of bacteria when the growth rate is known for specific time intervals.

The statement is False.

Explanation:

The first part of the statement is correct: additional expenses made after buying an article (such as transportation, labour charges, repair costs, etc.) are indeed included in the cost price to determine the total cost incurred by the buyer. These are often called overhead expenses.

However, the second part is incorrect. These additional expenses are not known as Value Added Tax (VAT).

Value Added Tax (VAT) is a consumption tax levied by the government on the sale of goods and services. It is typically calculated as a percentage of the value added at each stage of production or distribution, or in simple terms, as a percentage of the selling price paid by the consumer. VAT is added to the selling price, not included as an additional expense incurred by the buyer to modify or transport the product after purchase.

Therefore, while additional expenses are added to the cost price, they are different from Value Added Tax.

The statement is False.

Explanation:

Discount is a reduction given on the Marked Price (or List Price) of an article, not on the Cost Price.

The Cost Price is the price at which the seller purchased the article.

The Marked Price is the price labelled on the article by the seller, which is usually higher than the cost price.

Discount is offered on this Marked Price to attract customers and arrive at the Selling Price.

Selling Price = Marked Price - Discount

The discount is calculated as a percentage of the Marked Price:

Discount Amount $= \frac{\text{Discount Percentage}}{100} \times \text{Marked Price}$

Therefore, the statement that discount is given on the cost price is incorrect.

The statement is True.

Explanation:

In Simple Interest, the interest is calculated only on the original principal amount for the entire duration.

In Compound Interest, the interest earned in one period is added to the principal to form a new principal for the next period. The interest for the current period is then calculated on this new, increased principal amount.

The "previous year's amount" refers to the principal plus the interest accumulated up to the end of the previous year. Calculating interest on this amount is the fundamental principle of compounding.

For example, if you invest $\textsf{₹} P$ at $R\%$ annual interest, the amount at the end of Year 1 is $P(1 + R/100)$. The interest for Year 2 is calculated on this amount, $P(1 + R/100)$, not just on the original principal $P$.

Therefore, the statement correctly describes how compound interest works.

The statement is False.

Explanation:

The correct relationship involving Marked Price (M.P.) and Discount is with the Selling Price (S.P.).

The formula is:

Selling Price = Marked Price - Discount

The Cost Price (C.P.) is the price at which the shopkeeper buys the article. The Selling Price is the price at which the shopkeeper sells it. The relationship between C.P. and S.P. involves Profit or Loss:

If there is a Profit: S.P. = C.P. + Profit

If there is a Loss: S.P. = C.P. - Loss

There is no direct formula stating C.P. = M.P. - Discount. The C.P. and M.P. are different values, and the discount is a reduction from the M.P. to arrive at the S.P.

The statement is False.

Explanation:

Given:

Cost Price (CP) = $\textsf{₹}$ 1,040

Selling Price (SP) = $\textsf{₹}$ 800

To determine if there is a gain or loss, we compare the Selling Price and the Cost Price.

Since the Selling Price ($\textsf{₹}$ 800) is less than the Cost Price ($\textsf{₹}$ 1,040), there is a Loss in this transaction, not a gain.

The amount of loss is:

Loss = Cost Price - Selling Price

Loss $= \textsf{₹} 1040 - \textsf{₹} 800$

Loss $= \textsf{₹} 240$

Since there is a loss, the claim that his gain per cent is 30% is incorrect. There is a loss percentage, not a gain percentage.

(Note: If we were to calculate the loss percentage, it would be $\left(\frac{\text{Loss}}{\text{CP}}\right) \times 100\% = \left(\frac{240}{1040}\right) \times 100\% \approx 23.08\%$.)

Therefore, the statement is false.

The statement is False.

Explanation:

Let the original number be $x$ (assuming $x \neq 0$).

Part 1: "Three times a number is 200% increase in the number."

Three times the number is $3x$.

The increase from $x$ to $3x$ is $3x - x = 2x$.

The percentage increase is calculated relative to the original number ($x$).

Percentage Increase $= \left(\frac{\text{Increase}}{\text{Original Number}}\right) \times 100\%$

Percentage Increase $= \left(\frac{2x}{x}\right) \times 100\%$

Percentage Increase $= 2 \times 100\% = 200\%$.

This part of the statement is True.

Part 2: "one-third of the same number is 200% decrease in the number."

One-third of the number is $\frac{1}{3}x$.

Since $\frac{1}{3}x < x$ (for $x>0$), there is a decrease.

The decrease from $x$ to $\frac{1}{3}x$ is $x - \frac{1}{3}x = \frac{3x - x}{3} = \frac{2}{3}x$.

The percentage decrease is calculated relative to the original number ($x$).

Percentage Decrease $= \left(\frac{\text{Decrease}}{\text{Original Number}}\right) \times 100\%$

Percentage Decrease $= \left(\frac{\frac{2}{3}x}{x}\right) \times 100\%$

Percentage Decrease $= \frac{2}{3} \times 100\% = \frac{200}{3}\%$.

$\frac{200}{3}\% = 66\frac{2}{3}\%$.

This is not 200%. Therefore, this part of the statement is False.

Since the second part of the statement is false, the entire statement is false.

The statement is True.

Explanation:

Let the Principal be $P$, the annual rate of interest be $R\%$, and the time period be $t$ years.

The formula for Simple Interest (SI) is:

$SI = \frac{P \times R \times t}{100}$

The formula for the Amount ($A$) under Compound Interest (CI) compounded annually is:

$A = P \left(1 + \frac{R}{100}\right)^t$

The Compound Interest (CI) is then:

$CI = A - P = P \left(1 + \frac{R}{100}\right)^t - P$

Let's compare SI and CI for different time periods:

Case 1: For time period $t=1$ year

$SI = \frac{P \times R \times 1}{100} = \frac{PR}{100}$

$CI = P \left(1 + \frac{R}{100}\right)^1 - P = P \left(\frac{100+R}{100}\right) - P = \frac{P(100+R) - 100P}{100} = \frac{100P + PR - 100P}{100} = \frac{PR}{100}$

So, for $t=1$ year, $SI = CI$.

Case 2: For time period $t > 1$ year (assuming $P > 0$ and $R > 0$)

Consider the expansion of $\left(1 + \frac{R}{100}\right)^t$ using the binomial theorem or by simple comparison.

For $t > 1$, $\left(1 + \frac{R}{100}\right)^t > 1 + t \times \frac{R}{100}$.

Multiplying by $P$:

$P \left(1 + \frac{R}{100}\right)^t > P \left(1 + \frac{tR}{100}\right)$

$P \left(1 + \frac{R}{100}\right)^t > P + \frac{PRt}{100}$

Subtracting $P$ from both sides:

$P \left(1 + \frac{R}{100}\right)^t - P > \frac{PRt}{100}$

This shows that $CI > SI$ for $t > 1$.

In summary:

- If time period is 1 year, Simple Interest equals Compound Interest.

- If time period is greater than 1 year (and rate and principal are positive), Compound Interest is greater than Simple Interest because interest earned in earlier periods also earns interest.

- If Principal or Rate is 0, both SI and CI are 0, so they are equal.

Combining these cases, Simple Interest is always less than or equal to Compound Interest for the same principal, rate, and time period.

The statement is True.

Explanation:

Given:

Initial Cost of the machine = $\textsf{₹}$ 7,000

Depreciation Rate per annum (R%) = 8%

Time Period (n) = 2 years

To Verify:

The value of the machine after 2 years is $\textsf{₹}$ 5,924.80.

Solution:

The formula for the value of an article after depreciation is similar to the compound interest formula, but with a negative rate.

Value after $n$ years $= \text{Initial Value} \times \left(1 - \frac{\text{Rate}}{100}\right)^n$

Value after 2 years $= 7000 \times \left(1 - \frac{8}{100}\right)^2$

Value after 2 years $= 7000 \times \left(1 - 0.08\right)^2$

Value after 2 years $= 7000 \times (0.92)^2$

Let's calculate $(0.92)^2$:

$0.92 \times 0.92 = 0.8464$

Now, substitute this back into the formula:

Value after 2 years $= 7000 \times 0.8464$

$= 7 \times 1000 \times 0.8464$

$= 7 \times 846.4$

Calculate $7 \times 846.4$:

$846.4 \times 7 = 5924.8$

Value after 2 years $= \textsf{₹}$ 5924.80

The calculated value of the machine after 2 years is $\textsf{₹}$ 5,924.80.

This matches the value given in the statement.

Therefore, the statement is true.

The statement is False.

Explanation:

Given:

Discount Amount = $\textsf{₹}$ y

Marked Price (MP) = $\textsf{₹}$ x

The formula for calculating the Discount Percentage is always based on the Marked Price.

Discount Percentage $= \left(\frac{\text{Discount Amount}}{\text{Marked Price}}\right) \times 100\%$

Substituting the given values into the correct formula:

Discount Percentage $= \left(\frac{\text{y}}{\text{x}}\right) \times 100\%$

The formula given in the statement is $\frac{x}{y} \times 100\%$, which is the reciprocal of the correct formula.

Therefore, the statement is false.

The statement is True.

Explanation:

Given:

Number of students in 1999-2000 = 91,422

Number of students in 2008-09 = 116,054

To Verify:

The percentage increase is approximately 27%.

Solution:

First, calculate the absolute increase in the number of students.

Increase in number of students = Number in 2008-09 - Number in 1999-2000

Increase = $116054 - 91422$

Increase $= 24632$

Next, calculate the percentage increase. The percentage increase is calculated with respect to the initial number of students (in 1999-2000).

Percentage Increase $= \left(\frac{\text{Increase}}{\text{Initial Number}}\right) \times 100\%$

Percentage Increase $= \left(\frac{24632}{91422}\right) \times 100\%$

Let's perform the calculation:

$\frac{24632}{91422} \approx 0.26942$

Percentage Increase $\approx 0.26942 \times 100\%$

Percentage Increase $\approx 26.942\%$

The calculated percentage increase is approximately 26.94%. This is very close to 27%.

Since 26.94% is approximately 27%, the statement is true.

The statement is True.

Explanation:

Given:

Selling Price (SP) of 9 articles = Cost Price (CP) of 15 articles.

To Verify:

The profit percentage in this transaction is $66 \frac{2}{3}\%$.

Solution:

Let CP be the cost price of one article.

Let SP be the selling price of one article.

According to the given condition:

We need to find the profit percentage on the articles sold. In this case, 9 articles are sold.

Cost Price of 9 articles = $9 \times \text{CP}$

Selling Price of 9 articles = $9 \times \text{SP}$

From the given condition, we know that $9 \times \text{SP}$ is equal to the cost price of 15 articles.

Selling Price of 9 articles $= 15 \times \text{CP}$

Since the Selling Price of 9 articles ($\textsf{₹}$ corresponding to 15 CP) is greater than the Cost Price of 9 articles ($\textsf{₹}$ corresponding to 9 CP), there is a profit.

Profit on selling 9 articles = Selling Price of 9 articles - Cost Price of 9 articles

Profit on selling 9 articles $= (15 \times \text{CP}) - (9 \times \text{CP})$

Profit on selling 9 articles $= 6 \times \text{CP}$

The Profit Percentage is calculated on the Cost Price of the articles that were sold (which is 9 articles).

Profit \% $= \left(\frac{\text{Profit on 9 articles}}{\text{Cost Price of 9 articles}}\right) \times 100\%$

Profit \% $= \left(\frac{6 \times \text{CP}}{9 \times \text{CP}}\right) \times 100\%$

Assuming CP is not zero, we can cancel CP:

Profit \% $= \left(\frac{6}{9}\right) \times 100\%$

Simplify the fraction $\frac{6}{9}$ by dividing numerator and denominator by 3:

Profit \% $= \frac{2}{3} \times 100\%$

Profit \% $= \frac{200}{3}\%$

Convert the improper fraction to a mixed number:

$200 \div 3$

$200 = 3 \times 66 + 2$

So, $\frac{200}{3} = 66 \frac{2}{3}$.

Profit \% $= 66 \frac{2}{3}\%$

The calculated profit percentage is $66 \frac{2}{3}\%$, which matches the percentage given in the statement.

Therefore, the statement is true.

The statement is False.

Explanation:

Given Formula:

The formula provided in the statement for Compound Interest (CI) is:

CI $= P \left( 1+\frac{R}{100} \right)$

Correct Formula for Amount (A):

When a principal $P$ is compounded annually at a rate of $R\%$ per annum for $T$ years, the formula for the Amount ($A$) is:

$A = P \left(1 + \frac{R}{100}\right)^T$

Correct Formula for Compound Interest (CI):

The Compound Interest is the difference between the Amount and the Principal:

CI = Amount - Principal

CI $= P \left(1 + \frac{R}{100}\right)^T - P$

Comparing the given formula $P \left( 1+\frac{R}{100} \right)$ with the correct formulas:

The formula $P \left( 1+\frac{R}{100} \right)$ is actually the formula for the Amount when the time period $T$ is exactly 1 year.

$A_{\text{for 1 year}} = P \left(1 + \frac{R}{100}\right)^1 = P \left(1 + \frac{R}{100}\right)$

This is the Amount after one year, not the Compound Interest for T years.

The correct formula for Compound Interest for $T$ years is $P \left(1 + \frac{R}{100}\right)^T - P$.

Since the formula provided in the statement does not give the compound interest for $T$ years (unless $T=1$ and it's mistaken for CI instead of Amount), the statement is false.

The statement is True.

Explanation:

Let C.P. be the Cost Price and S.P. be the Selling Price of an article.

When there is a gain, it means S.P. > C.P.

The amount of Gain is given by:

Gain = S.P. - C.P.

The Gain Percentage (Gain%) is calculated on the Cost Price:

Gain \% $= \left(\frac{\text{Gain}}{\text{C.P.}}\right) \times 100$

From the Gain% formula, we can express Gain in terms of C.P. and Gain%:

Gain $= \frac{\text{Gain}\%}{100} \times \text{C.P.}$

Now, substitute this expression for Gain into the formula S.P. = C.P. + Gain:

S.P. $= \text{C.P.} + \left(\frac{\text{Gain}\%}{100} \times \text{C.P.}\right)$

Take C.P. as a common factor:

S.P. $= \text{C.P.} \left(1 + \frac{\text{Gain}\%}{100}\right)$

Combine the terms inside the parenthesis by finding a common denominator:

S.P. $= \text{C.P.} \left(\frac{100}{100} + \frac{\text{Gain}\%}{100}\right)$

S.P. $= \text{C.P.} \left(\frac{100 + \text{Gain}\%}{100}\right)$

Rewrite the multiplication:

S.P. $= \frac{(100 + \text{Gain}\%) \times \text{C.P.}}{100}$

This derived formula matches the formula given in the statement exactly.

Therefore, the statement is true.

The statement is False.

Explanation:

Let C.P. be the Cost Price and S.P. be the Selling Price of an article.

When there is a loss, it means S.P. < C.P.

The amount of Loss is given by:

Loss = C.P. - S.P.

The Loss Percentage (Loss%) is calculated on the Cost Price:

Loss \% $= \left(\frac{\text{Loss}}{\text{C.P.}}\right) \times 100$

From the Loss% formula, we can express Loss in terms of C.P. and Loss%:

Loss $= \frac{\text{Loss}\%}{100} \times \text{C.P.}$

Now, substitute this expression for Loss into the formula Loss = C.P. - S.P.:

$\frac{\text{Loss}\%}{100} \times \text{C.P.} = \text{C.P.} - \text{S.P.}$

Rearrange the equation to isolate S.P.:

S.P. $= \text{C.P.} - \left(\frac{\text{Loss}\%}{100} \times \text{C.P.}\right)$

Factor out C.P.:

S.P. $= \text{C.P.} \left(1 - \frac{\text{Loss}\%}{100}\right)$

Combine terms inside the parenthesis:

S.P. $= \text{C.P.} \left(\frac{100 - \text{Loss}\%}{100}\right)$

Now, solve this equation for C.P.:

C.P. $= \text{S.P.} \times \left(\frac{100}{100 - \text{Loss}\%}\right)$

C.P. $= \frac{100 \times \text{S.P.}}{100 - \text{Loss}\%}$

This is the correct formula for C.P. in case of loss. The formula given in the statement has $100 + \text{Loss}\%$ in the denominator instead of $100 - \text{Loss}\%$.

Therefore, the statement is false.

The statement is False.

Explanation:

Given:

Initial Value of the car (P) = $\textsf{₹}$ 4,40,000

Depreciation Rate per annum (R%) = 10%

Time Period (n) = 3 years

To Verify:

The value of the car after 3 years is $\textsf{₹}$ 3,08,000.

Solution:

When an article depreciates at a constant rate each year, the formula for its value after $n$ years is:

Value after $n$ years $= \text{Initial Value} \times \left(1 - \frac{\text{Rate}}{100}\right)^n$

Substitute the given values:

Value after 3 years $= 440000 \times \left(1 - \frac{10}{100}\right)^3$

Value after 3 years $= 440000 \times \left(1 - 0.1\right)^3$

Value after 3 years $= 440000 \times (0.9)^3$

Calculate $(0.9)^3$:

$(0.9)^3 = 0.9 \times 0.9 \times 0.9 = 0.81 \times 0.9 = 0.729$

Now, calculate the value after 3 years:

Value after 3 years $= 440000 \times 0.729$

Value after 3 years $= 440000 \times \frac{729}{1000}$

Value after 3 years $= 440 \times 729$

$\begin{array}{cc}& & 7 & 2 & 9 \\ \times & & 4 & 4 & 0 \\ \hline && 0 & 0 & 0 \\ & 2 & 9 & 1 & 6 & \times \\ 2 & 9 & 1 & 6 & \times & \times \\ \hline 3 & 2 & 0 & 7 & 6 & 0 \\ \hline \end{array}$

Value after 3 years $= \textsf{₹}$ 3,20,760

The calculated value of the car after 3 years is $\textsf{₹}$ 3,20,760.

The value given in the statement is $\textsf{₹}$ 3,08,000.

Since $\textsf{₹} 3,20,760 \neq \textsf{₹} 3,08,000$, the statement is false.

The statement is False.

Explanation:

Given:

Marked Price (MP) = $\textsf{₹}$ 190

Sales Tax Rate = 2%

To Verify:

The total cost (inclusive of sales tax) is $\textsf{₹}$ 192.

Solution:

Assuming the Selling Price (SP) is equal to the Marked Price (MP) since no discount is mentioned:

Selling Price (SP) = $\textsf{₹}$ 190

The sales tax is charged on the Selling Price.

Sales Tax Amount $= 2\%$ of Selling Price

Sales Tax Amount $= \frac{2}{100} \times 190$

Sales Tax Amount $= \frac{380}{100}$

Sales Tax Amount $= \textsf{₹} 3.80$

The total amount paid is the Selling Price plus the Sales Tax.

Total Amount Paid = Selling Price + Sales Tax Amount

Total Amount Paid $= \textsf{₹} 190 + \textsf{₹} 3.80$

Total Amount Paid $= \textsf{₹} 193.80$

The calculated total amount paid is $\textsf{₹}$ 193.80. The statement claims it is $\textsf{₹}$ 192.

Since $\textsf{₹} 193.80 \neq \textsf{₹} 192$, the statement is false.

The statement is False.

Explanation:

Given:

Quantity of flour = 5 kg

Rate per kg = $\textsf{₹}$ 20

Sales Tax (ST) Rate = 5%

To Verify:

The total buying price (inclusive of sales tax) is $\textsf{₹}$ 21.

Solution:

First, calculate the basic cost of 5 kg of flour before adding sales tax.

Basic Cost = Quantity $\times$ Rate per kg

Basic Cost $= 5 \text{ kg} \times \textsf{₹} 20/\text{kg}$

Basic Cost $= \textsf{₹} 100$

Next, calculate the amount of sales tax on this basic cost.

Sales Tax Amount $= 5\%$ of Basic Cost

Sales Tax Amount $= \frac{5}{100} \times 100$

Sales Tax Amount $= \frac{500}{100}$

Sales Tax Amount $= \textsf{₹} 5$

Finally, calculate the total buying price by adding the sales tax to the basic cost.

Total Buying Price = Basic Cost + Sales Tax Amount

Total Buying Price $= \textsf{₹} 100 + \textsf{₹} 5$

Total Buying Price $= \textsf{₹} 105$

The calculated total buying price is $\textsf{₹}$ 105. The statement claims it is $\textsf{₹}$ 21.

Since $\textsf{₹} 105 \neq \textsf{₹} 21$, the statement is false.

It seems there might be a misunderstanding in the question or the proposed answer. If the question meant the cost of 1 kg was $\textsf{₹}$ 20, then the cost would be $\textsf{₹} 20 + 5\%$ of $\textsf{₹} 20 = \textsf{₹} 20 + \textsf{₹} 1 = \textsf{₹} 21$. However, the question clearly states "buying price of 5 kg of flour".

The statement is True.

Explanation:

Given:

Price of the shampoo bottle inclusive of VAT = $\textsf{₹}$ 324

VAT Rate = 8%

To Verify:

The original price (price before VAT) is $\textsf{₹}$ 300.

Solution:

Let the original price of the shampoo bottle (before VAT) be $\textsf{₹} P$.

VAT is charged at 8% on the original price.

VAT Amount $= 8\%$ of $P$

VAT Amount $= \frac{8}{100} \times P = 0.08 P$

The price inclusive of VAT is the original price plus the VAT amount.

Price inclusive of VAT = Original Price + VAT Amount

$324 = P + 0.08 P$

$324 = P (1 + 0.08)$

$324 = P (1.08)$

To find the original price ($P$), we divide the inclusive price by 1.08.

$P = \frac{324}{1.08}$

To perform the division, remove the decimal by multiplying the numerator and denominator by 100:

$P = \frac{324 \times 100}{1.08 \times 100}$

$P = \frac{32400}{108}$

Now, perform the division $32400 \div 108$. We know that $108 \times 3 = 324$.

$P = \frac{\cancel{324}^{3}00}{\cancel{108}_{1}}$ (Since $324 = 3 \times 108$)

$P = 3 \times 100$

$P = 300$

The calculated original price is $\textsf{₹}$ 300. This matches the price given in the statement.

We can verify this:

VAT on $\textsf{₹}$ 300 at 8% $= \frac{8}{100} \times 300 = \textsf{₹} 24$.

Price inclusive of VAT $= \textsf{₹} 300 + \textsf{₹} 24 = \textsf{₹} 324$.

This matches the given inclusive price.

Therefore, the statement is true.

The statement is False.

Explanation:

The statement consists of two parts connected by "and":

1. Sales tax is always calculated on the cost price of an item.

2. Sales tax is added to the value of the bill.

Let's examine the first part. Sales tax is a tax on the sale of goods and services. It is typically calculated as a percentage of the Selling Price (or sometimes the Marked Price) of the item, which is the price the customer is supposed to pay for the item itself before tax. The Cost Price is the price at which the seller originally bought the item. Sales tax is not calculated on the seller's cost price. So, the first part of the statement is false.

The second part states that sales tax is added to the value of the bill. This is true. The calculated sales tax amount is added to the selling price of the item to arrive at the final amount the customer has to pay, which is reflected in the bill.

Total Bill Amount = Selling Price + Sales Tax Amount

Since the first part of the statement is false (sales tax is calculated on Selling Price, not Cost Price), the entire statement is false, even though the second part is true.

Given:

Percentage of women workers = $35\%$.

Since the rest are men, percentage of men workers = $(100 - 35)\% = 65\%$.

The number of men exceeds the number of women by $252$.

To Find:

The total number of workers in the factory.

Solution:

Let the total number of workers in the factory be $W$.

Number of women workers = $35\%$ of $W = 0.35W$.

Number of men workers = $65\%$ of $W = 0.65W$.

According to the given information, the difference between the number of men and women is $252$.

Number of men - Number of women = $252$

$0.65W - 0.35W = 252$

Combining the terms with $W$:

$(0.65 - 0.35)W = 252$

$0.30W = 252$

To find $W$, divide both sides by $0.30$:

$W = \frac{252}{0.30}$

$W = \frac{252}{\frac{30}{100}}$

$W = 252 \times \frac{100}{30}$

$W = \frac{252 \times \cancel{100}^{10}}{\cancel{30}_{3}}$

$W = \frac{\cancel{252}^{84} \times 10}{\cancel{3}_{1}}$

$W = 84 \times 10$

$W = 840$

Verification:

Total workers = $840$.

Number of women = $35\%$ of $840 = 0.35 \times 840 = 294$.

Number of men = $65\%$ of $840 = 0.65 \times 840 = 546$.

Difference between men and women = $546 - 294 = 252$.

This matches the given condition.

Final Answer:

The total number of workers in the factory is $840$.

Given:

Total quantity of sugar in three bags = $64.2$ kg.

Quantity in the second bag is $\frac{4}{5}$ of the quantity in the first bag.

Quantity in the third bag is $45\frac{1}{2}\%$ of the quantity in the second bag.

To Find:

The quantity of sugar in each bag.

Solution:

Let the quantity of sugar in the first bag be $B_1$ kg.

According to the problem, the quantity of sugar in the second bag, $B_2$, is $\frac{4}{5}$ of $B_1$.

$B_2 = \frac{4}{5} B_1$

$B_2 = 0.8 B_1$

The quantity of sugar in the third bag, $B_3$, is $45\frac{1}{2}\%$ of $B_2$.

$45\frac{1}{2}\% = 45.5\% = \frac{45.5}{100} = \frac{455}{1000} = \frac{91}{200}$.

So, $B_3 = \frac{91}{200} B_2$.

Substitute the expression for $B_2$ in terms of $B_1$ into the equation for $B_3$:

$B_3 = \frac{91}{200} \left(\frac{4}{5} B_1\right)$

$B_3 = \frac{91 \times 4}{200 \times 5} B_1$

$B_3 = \frac{364}{1000} B_1$

$B_3 = 0.364 B_1$

The total quantity of sugar in the three bags is $B_1 + B_2 + B_3 = 64.2$ kg.

Substitute the expressions for $B_2$ and $B_3$ in terms of $B_1$ into the total sum equation:

$B_1 + 0.8 B_1 + 0.364 B_1 = 64.2$

Combine the terms with $B_1$:

$(1 + 0.8 + 0.364) B_1 = 64.2$

$2.164 B_1 = 64.2$

To find $B_1$, divide $64.2$ by $2.164$:

$B_1 = \frac{64.2}{2.164}$

To remove the decimal from the denominator, multiply the numerator and denominator by $1000$:

$B_1 = \frac{64.2 \times 1000}{2.164 \times 1000} = \frac{64200}{2164}$

Simplify the fraction by dividing the numerator and denominator by their greatest common divisor. Both are divisible by $4$:

$B_1 = \frac{\cancel{64200}^{16050}}{\cancel{2164}_{541}}$

$B_1 = \frac{16050}{541}$ kg.

Now find $B_2$ and $B_3$ using this value of $B_1$.

$B_2 = \frac{4}{5} B_1 = \frac{4}{5} \times \frac{16050}{541}$

$B_2 = \frac{4 \times \cancel{16050}^{3210}}{5 \times 541}$

$B_2 = \frac{4 \times 3210}{541} = \frac{12840}{541}$ kg.

$B_3 = \frac{91}{200} B_2 = \frac{91}{200} \times \frac{12840}{541}$

$B_3 = \frac{91 \times \cancel{12840}^{64.2}}{\cancel{200}_{1} \times 541}$ ... this is incorrect cancellation. Let's use the fraction form directly.

$B_3 = \frac{91 \times 12840}{200 \times 541}$

$B_3 = \frac{1168440}{108200}$

Simplify the fraction:

$B_3 = \frac{\cancel{1168440}^{58422}}{\cancel{108200}_{5410}}$ (Dividing by 20)

$B_3 = \frac{58422}{5410}$ kg.

Alternatively, using $B_3 = 0.364 B_1$:

$B_3 = 0.364 \times \frac{16050}{541} = \frac{364}{1000} \times \frac{16050}{541}$

$B_3 = \frac{364 \times 16050}{1000 \times 541} = \frac{5842200}{541000}$

Simplify the fraction:

$B_3 = \frac{\cancel{5842200}^{58422}}{\cancel{541000}_{5410}}$ (Dividing by 100)

$B_3 = \frac{58422}{5410}$ kg.

Final Answer:

The quantity of sugar in each bag is:

First bag: $\mathbf{\frac{16050}{541}}$ kg

Second bag: $\mathbf{\frac{12840}{541}}$ kg

Third bag: $\mathbf{\frac{58422}{5410}}$ kg

(Note: These are exact values. Decimal approximations would be approximately 29.67 kg, 23.73 kg, and 10.80 kg respectively, summing to approximately 64.2 kg).

We need to find the Selling Price (S.P.) using the formula:

S.P. = M.P. - Discount Amount

where, Discount Amount = Discount Percentage $\times$ M.P.

(a)

Given:

M.P. = $\textsf{₹}$ $5450$

Discount = $5\%$

Calculation:

Discount Amount = $5\%$ of $\textsf{₹}$ $5450$

Discount Amount = $\frac{5}{100} \times 5450$

Discount Amount = $0.05 \times 5450$

Discount Amount = $\textsf{₹}$ $272.50$

S.P. = M.P. - Discount Amount

S.P. = $5450 - 272.50$

S.P. = $\textsf{₹}$ $5177.50$

Final Answer for (a):

The S.P. is $\mathbf{\textsf{₹} \ 5177.50}$.

(b)

Given:

M.P. = $\textsf{₹}$ $1300$

Discount = $1.5\%$

Calculation:

Discount Amount = $1.5\%$ of $\textsf{₹}$ $1300$

Discount Amount = $\frac{1.5}{100} \times 1300$

Discount Amount = $0.015 \times 1300$

Discount Amount = $\textsf{₹}$ $19.50$

S.P. = M.P. - Discount Amount

S.P. = $1300 - 19.50$

S.P. = $\textsf{₹}$ $1280.50$

Final Answer for (b):

The S.P. is $\mathbf{\textsf{₹} \ 1280.50}$.

We can find the Marked Price (M.P.) using the relationship between Selling Price (S.P.), M.P., and Discount Percentage.

S.P. = M.P. - Discount

Discount = Discount Percentage $\times$ M.P.

So, S.P. = M.P. - (Discount Percentage $\times$ M.P.)

S.P. = M.P. $\times$ (1 - Discount Rate)

Therefore, M.P. = $\frac{\text{S.P.}}{1 - \text{Discount Rate}}$

(a)

Given:

S.P. = $\textsf{₹}$ $495$

Discount = $1\%$

Calculation:

Discount Rate = $1\% = \frac{1}{100} = 0.01$

M.P. = $\frac{495}{1 - 0.01}$

M.P. = $\frac{495}{0.99}$

M.P. = $\frac{495}{\frac{99}{100}}$

M.P. = $495 \times \frac{100}{99}$

M.P. = $\frac{\cancel{495}^{5} \times 100}{\cancel{99}_{1}}$

M.P. = $5 \times 100$

M.P. = $\textsf{₹}$ $500$

Final Answer for (a):

The M.P. is $\mathbf{\textsf{₹} \ 500}$.

(b)

Given:

S.P. = $\textsf{₹}$ $9,250$

Discount = $7\frac{1}{2}$ %

Calculation:

Discount Percentage = $7\frac{1}{2}\% = 7.5\% = \frac{7.5}{100} = 0.075$

M.P. = $\frac{9250}{1 - 0.075}$

M.P. = $\frac{9250}{0.925}$

M.P. = $\frac{9250}{\frac{925}{1000}}$

M.P. = $9250 \times \frac{1000}{925}$

M.P. = $\frac{\cancel{9250}^{10} \times 1000}{\cancel{925}_{1}}$

M.P. = $10 \times 1000$

M.P. = $\textsf{₹}$ $10,000$

Final Answer for (b):

The M.P. is $\mathbf{\textsf{₹} \ 10,000}$.

We can find the discount in per cent using the formula:

Discount Percentage = $\frac{\text{Discount Amount}}{\text{M.P.}} \times 100\%$

where, Discount Amount = M.P. - S.P.

(a)

Given:

M.P. = $\textsf{₹}$ $625$

S.P. = $\textsf{₹}$ $562.50$

Calculation:

Discount Amount = M.P. - S.P.

Discount Amount = $625 - 562.50$

Discount Amount = $\textsf{₹}$ $62.50$

Discount Percentage = $\frac{62.50}{625} \times 100\%$

Discount Percentage = $\frac{62.5}{625} \times 100\%$

Discount Percentage = $\frac{625}{6250} \times 100\%$

Discount Percentage = $\frac{1}{10} \times 100\%$

Discount Percentage = $10\%$

Final Answer for (a):

The discount percentage is $\mathbf{10\%}$.

(b)

Given:

M.P. = $\textsf{₹}$ $900$

S.P. = $\textsf{₹}$ $873$

Calculation:

Discount Amount = M.P. - S.P.

Discount Amount = $900 - 873$

Discount Amount = $\textsf{₹}$ $27$

Discount Percentage = $\frac{27}{900} \times 100\%$

Discount Percentage = $\frac{27}{\cancel{900}_{9}} \times \cancel{100}^{1}\%$

Discount Percentage = $\frac{\cancel{27}^{3}}{\cancel{9}_{1}}\%$

Discount Percentage = $3\%$

Final Answer for (b):

The discount percentage is $\mathbf{3\%}$.

Given:

Marked Price (M.P.) = $\textsf{₹}$ $500$.

Discount = $5\%$.

Profit = $25\%$.

To Find:

The Cost Price (C.P.) of the article.

Solution:

First, we calculate the Selling Price (S.P.) after the discount.

Discount Amount = Discount Percentage $\times$ M.P.

Discount Amount = $5\%$ of $\textsf{₹}$ $500$

Discount Amount = $\frac{5}{100} \times 500$

Discount Amount = $5 \times 5 = \textsf{₹} \ 25$

Selling Price (S.P.) = M.P. - Discount Amount

S.P. = $500 - 25$

S.P. = $\textsf{₹} \ 475$

Now, we use the S.P. and the Profit percentage to find the C.P.

S.P. = C.P. + Profit

Profit = Profit Percentage $\times$ C.P.

So, S.P. = C.P. + ($25\%$ of C.P.)

S.P. = C.P. + $\frac{25}{100} \times$ C.P.

S.P. = C.P. + $0.25 \times$ C.P.

S.P. = $(1 + 0.25) \times$ C.P.

S.P. = $1.25 \times$ C.P.

Substitute the calculated S.P. value:

$475 = 1.25 \times$ C.P.

C.P. = $\frac{475}{1.25}$

To remove the decimal, multiply the numerator and denominator by $100$:

C.P. = $\frac{475 \times 100}{1.25 \times 100}$

C.P. = $\frac{47500}{125}$

Simplify the fraction:

C.P. = $\frac{\cancel{47500}^{9500}}{\cancel{125}_{25}}$ (Dividing by 5)

C.P. = $\frac{\cancel{9500}^{1900}}{\cancel{25}_{5}}$ (Dividing by 5)

C.P. = $\frac{\cancel{1900}^{380}}{\cancel{5}_{1}}$ (Dividing by 5)

C.P. = $380$

The cost price of the article is $\textsf{₹}$ $380$.

Final Answer:

The cost price of the article is $\mathbf{\textsf{₹} \ 380}$.

Given:

For the year 2007-08:

Number of students appeared = $1,05,332$

Number of students passed = $88,151$

For the year 2008-09:

Number of students appeared = $1,16,054$

Number of students passed = $1,03,804$

To Find:

The increase or decrease in pass percentage between the two years.

Solution:

First, calculate the pass percentage for the year 2007-08.

Pass Percentage (2007-08) = $\frac{\text{Number of students passed}}{\text{Number of students appeared}} \times 100\%$

Pass Percentage (2007-08) = $\frac{88151}{105332} \times 100\%$

Pass Percentage (2007-08) $\approx 0.836894 \times 100\%$

Pass Percentage (2007-08) $\approx 83.69\%$ (rounded to two decimal places)

Next, calculate the pass percentage for the year 2008-09.

Pass Percentage (2008-09) = $\frac{\text{Number of students passed}}{\text{Number of students appeared}} \times 100\%$

Pass Percentage (2008-09) = $\frac{103804}{116054} \times 100\%$

Pass Percentage (2008-09) $\approx 0.894445 \times 100\%$

Pass Percentage (2008-09) $\approx 89.44\%$ (rounded to two decimal places)

Now, compare the pass percentages for the two years to find the increase or decrease.

Change in Pass Percentage = Pass Percentage (2008-09) - Pass Percentage (2007-08)

Change in Pass Percentage $\approx 89.44\% - 83.69\%$

Change in Pass Percentage $\approx 5.75\%$

Since the result is positive, there is an increase in the pass percentage.

Final Answer:

The pass percentage increased from approximately $83.69\%$ in 2007-08 to approximately $89.44\%$ in 2008-09.

The increase in pass percentage is approximately $\mathbf{5.75\%}$.

Given:

Marked Price (M.P.) of the watch = $\textsf{₹}$ $5400$.

Selling Price (S.P.) of the watch = $\textsf{₹}$ $4500$.

To Find:

The percentage discount offered.

Solution:

First, we calculate the discount amount offered.

Discount Amount = M.P. - S.P.

Discount Amount = $5400 - 4500$

Discount Amount = $\textsf{₹}$ $900$.

Now, we calculate the discount percentage based on the Marked Price.

Discount Percentage = $\frac{\text{Discount Amount}}{\text{M.P.}} \times 100\%$

Discount Percentage = $\frac{900}{5400} \times 100\%$

Discount Percentage = $\frac{\cancel{900}^{1}}{\cancel{5400}_{6}} \times 100\%$

Discount Percentage = $\frac{1}{6} \times 100\%$

Discount Percentage = $\frac{100}{6}\%$

Discount Percentage = $\frac{50}{3}\% = 16\frac{2}{3}\%$

As a decimal, this is approximately $16.67\%$.

Final Answer:

The percentage discount offered during the sale is $\mathbf{16\frac{2}{3}\%}$ or approximately $\mathbf{16.67\%}$.

Given:

Number of malaria patients in 2001 = $4375$.

Annual decrease rate = $8\%$.

To Find:

The number of patients in 2003.

Solution:

The number of patients decreases by a fixed percentage each year. This is a case of compound decrease.

The number of years from 2001 to 2003 is $2003 - 2001 = 2$ years.

Let $P_0$ be the initial number of patients (in 2001), $r$ be the annual decrease rate, and $n$ be the number of years.

The number of patients after $n$ years, $P_n$, can be calculated using the formula:

$P_n = P_0 (1 - r)^n$

Here, $P_0 = 4375$, $r = 8\% = \frac{8}{100} = 0.08$, and $n = 2$.

Number of patients in 2003 = $4375 (1 - 0.08)^2$

Number of patients in 2003 = $4375 (0.92)^2$

Calculate $(0.92)^2$:

$(0.92)^2 = 0.92 \times 0.92$

$\begin{array}{cc}& & 0 & . & 9 & 2 \\ \times & & 0 & . & 9 & 2 \\ \hline &&& 1 & 8 & 4 \\ && 8 & 2 & 8 & \times \\ \hline && 8 & 4 & 6 & 4 \\ \hline \end{array}$

So, $(0.92)^2 = 0.8464$.

Number of patients in 2003 = $4375 \times 0.8464$

$\begin{array}{cc}& & 4 & 3 & 7 & 5 \\ \times & & 0 & . & 8 & 4 & 6 & 4 \\ \hline && 1 & 7 & 5 & 0 & 0 \\ & 2 & 6 & 2 & 5 & 0 & \times \\ & 1 & 7 & 5 & 0 & 0 & \times & \times \\ 3 & 5 & 0 & 0 & 0 & \times & \times & \times \\ \hline 3 & 7 & 0 & 3 & . & 0 & 0 & 0 & 0 \\ \hline \end{array}$

Number of patients in 2003 = $3703$.

Final Answer:

The number of malaria patients in 2003 was $\mathbf{3703}$.

Given:

Total price of the product (including sales tax) = $\textsf{₹}$ $3155$.

Sales tax percentage = $4.5\%$.

To Find:

The price of the product before tax was added.

Solution:

Let the price of the product before tax be $P$.

The sales tax is calculated on the original price $P$.

Sales Tax Amount = Sales Tax Percentage $\times$ Original Price

Sales Tax Amount = $4.5\%$ of $P$

Sales Tax Amount = $\frac{4.5}{100} \times P = 0.045 P$

The total price is the sum of the original price and the sales tax amount.

Total Price = Original Price + Sales Tax Amount

$3155 = P + 0.045 P$

Combine the terms with $P$:

$3155 = (1 + 0.045) P$

$3155 = 1.045 P$

To find $P$, divide the total price by $1.045$:

$P = \frac{3155}{1.045}$

To remove the decimal from the denominator, multiply the numerator and denominator by $1000$:

$P = \frac{3155 \times 1000}{1.045 \times 1000} = \frac{3155000}{1045}$

Simplify the fraction by dividing the numerator and denominator by their greatest common divisor. Both are divisible by $5$:

$P = \frac{\cancel{3155000}^{631000}}{\cancel{1045}_{209}}$

$P = \frac{631000}{209}$

Now, we can perform the division. Note that $209 \times 3 = 627$.

$631000 \div 209 = \frac{627000 + 4000}{209} = \frac{627000}{209} + \frac{4000}{209}$

$\frac{627000}{209} = 3000$.

So, $P = 3000$.

Verification:

Original Price = $\textsf{₹}$ $3000$.

Sales Tax Amount = $4.5\%$ of $\textsf{₹}$ $3000$

Sales Tax Amount = $\frac{4.5}{100} \times 3000 = 4.5 \times 30 = \textsf{₹} \ 135$.

Total Price = Original Price + Sales Tax Amount

Total Price = $3000 + 135 = \textsf{₹} \ 3135$.

Wait, this does not match the given total price of $\textsf{₹} \ 3155$. Let's recheck the division.

$P = \frac{3155000}{1045}$

Let's try dividing $3155$ by $1.045$.

$3155 / 1.045 \approx 3019.13875...$

It seems the number in the question might lead to a non-integer answer for the price before tax. Let's assume the calculation steps are correct based on the formula and the numbers provided.

Using the division: $\frac{3155000}{1045}$.

Divide 3155 by 1045: $3155 \div 1045 = 3$ with a remainder of $3155 - (3 \times 1045) = 3155 - 3135 = 20$.

So $3155/1045$ is not a clean division.

Let's re-read the question carefully. "Jyotsana bought a product for Rs 3,155 including 4.5% sales tax". This implies 3155 IS the final price with tax.

Our equation $3155 = 1.045 P$ is correct.

$P = \frac{3155}{1.045}$

Let's simplify the fraction $\frac{3155}{1045}$. Both are divisible by 5. $3155 \div 5 = 631$. $1045 \div 5 = 209$.

$P = \frac{631}{209} \times 1000 = \frac{631000}{209}$

Now let's try dividing $631$ by $209$. $631 = 3 \times 209 + 4$. $631 = 627 + 4$.

So $\frac{631}{209}$ is not a whole number. Perhaps the question expects a fractional or decimal answer.

$P = \frac{631000}{209}$

Let's perform long division for $631000 \div 209$ or check if 631 or 209 have other factors. 209 = 11 * 19. 631 is not divisible by 11 (631 = 57 * 11 + 4) or 19 (631 = 33 * 19 + 4).

It seems the exact answer is $\frac{631000}{209}$. Let's express this as a mixed number or decimal if preferred, but the question doesn't specify. Let's provide the exact fraction.

Price before tax = $\frac{631000}{209}$ $\textsf{₹}$

As a decimal approximation: $\frac{631000}{209} \approx 3019.138756...$

Let's re-read again to ensure no misinterpretation. No, the wording is clear. The total price *including* tax is 3155.

Let's re-calculate the verification step with the approximate value.

If P $\approx 3019.138756$, Sales tax $\approx 0.045 \times 3019.138756 \approx 135.861244$.

Original + Tax $\approx 3019.138756 + 135.861244 = 3155$. This confirms our method and the fraction $\frac{631000}{209}$ is correct for the given numbers.

Final Answer:

The price of the product before tax was added is $\mathbf{\textsf{₹} \ \frac{631000}{209}}$.

Given:

Total water usage per person per day = $150$ litres.

Water usage for each activity is provided in the table:

• Drinking: $3$ litres
• Cooking: $4$ litres
• Bathing: $20$ litres
• Sanitation: $40$ litres
• Washing clothes: $40$ litres
• Washing utensils: $20$ litres
• Gardening: $23$ litres

To Find:

(a) The percentage of water used for bathing and sanitation together.

(b) The percentage less water used for cooking compared to bathing.

(c) The percentage of water used for drinking, cooking, and gardening together.

Solution:

Total water used per day = $150$ litres.

(a) Percentage of water used for bathing and sanitation together:

Water used for bathing = $20$ litres.

Water used for sanitation = $40$ litres.

Total water used for bathing and sanitation = $20 + 40 = 60$ litres.

Percentage = $\frac{\text{Water used for bathing and sanitation}}{\text{Total water used}} \times 100\%$

Percentage = $\frac{60}{150} \times 100\%$

Percentage = $\frac{\cancel{60}^{2}}{\cancel{150}_{5}} \times 100\%$

Percentage = $\frac{2}{5} \times 100\%$

Percentage = $2 \times \frac{\cancel{100}^{20}}{\cancel{5}_{1}}\%$

Percentage = $2 \times 20\% = 40\%$.

(b) Percentage less water used for cooking in comparison to that used for bathing:

Water used for cooking = $4$ litres.

Water used for bathing = $20$ litres.

Difference in water usage = Water used for bathing - Water used for cooking

Difference = $20 - 4 = 16$ litres.

The percentage less is calculated with respect to the water used for bathing.

Percentage Less = $\frac{\text{Difference in water usage}}{\text{Water used for bathing}} \times 100\%$

Percentage Less = $\frac{16}{20} \times 100\%$

Percentage Less = $\frac{\cancel{16}^{4}}{\cancel{20}_{5}} \times 100\%$

Percentage Less = $\frac{4}{5} \times 100\%$

Percentage Less = $4 \times \frac{\cancel{100}^{20}}{\cancel{5}_{1}}\%$

Percentage Less = $4 \times 20\% = 80\%$.

(c) Percentage of water used for drinking, cooking and gardening together:

Water used for drinking = $3$ litres.

Water used for cooking = $4$ litres.

Water used for gardening = $23$ litres.

Total water used for drinking, cooking and gardening = $3 + 4 + 23 = 30$ litres.

Percentage = $\frac{\text{Water used for drinking, cooking, gardening}}{\text{Total water used}} \times 100\%$

Percentage = $\frac{30}{150} \times 100\%$

Percentage = $\frac{\cancel{30}^{1}}{\cancel{150}_{5}} \times 100\%$

Percentage = $\frac{1}{5} \times 100\%$

Percentage = $\frac{\cancel{100}^{20}}{\cancel{5}_{1}}\%$

Percentage = $20\%$.

Final Answer:

(a) Water used for bathing and sanitation together is $\mathbf{40\%}$ of the total water used.

(b) Water used for cooking is $\mathbf{80\%}$ less than that used for bathing.

(c) Water used for drinking, cooking and gardening together is $\mathbf{20\%}$ of the total water used.

Given:

Water consumption in 1975 = $3850$ cu.km/year.

Water consumption in 2000 = $6000$ cu.km/year.

Time period = $2000 - 1975 = 25$ years.

To Find:

The total percentage increase in consumption from 1975 to 2000.

The annual percentage increase in consumption (assuming uniform increase).

Solution:

First, we calculate the total increase in water consumption from 1975 to 2000.

Increase in consumption = Consumption in 2000 - Consumption in 1975

Increase in consumption = $6000 - 3850 = 2150$ cu.km/year.

Now, we calculate the total percentage increase based on the consumption in 1975.

Total Percentage Increase = $\frac{\text{Increase in consumption}}{\text{Consumption in 1975}} \times 100\%$

Total Percentage Increase = $\frac{2150}{3850} \times 100\%$

Total Percentage Increase = $\frac{215}{385} \times 100\%$

Simplify the fraction $\frac{215}{385}$ by dividing both numerator and denominator by $5$:

$\frac{\cancel{215}^{43}}{\cancel{385}_{77}}$

Total Percentage Increase = $\frac{43}{77} \times 100\% = \frac{4300}{77}\%$.

As a decimal, $\frac{4300}{77}\% \approx 55.84\%$.

Next, we find the annual percentage increase, assuming the consumption increases uniformly over the $25$ years.

This can be interpreted as the total percentage increase divided equally over the number of years.

Annual Percentage Increase = $\frac{\text{Total Percentage Increase}}{\text{Number of years}}$

Annual Percentage Increase = $\frac{\frac{4300}{77}\%}{25}$

Annual Percentage Increase = $\frac{4300}{77 \times 25}\%$

Annual Percentage Increase = $\frac{\cancel{4300}^{172}}{77 \times \cancel{25}_{1}}\%$

Annual Percentage Increase = $\frac{172}{77}\%$.

As a mixed number, $\frac{172}{77} = 2\frac{18}{77}\%$.

As a decimal, $\frac{172}{77}\% \approx 2.2338\%$.

Final Answer:

The total percentage increase in the consumption of water from 1975 to 2000 is $\mathbf{\frac{4300}{77}\%}$ or approximately $\mathbf{55.84\%}$.

The annual percentage increase in consumption (assuming uniform increase) is $\mathbf{\frac{172}{77}\%}$ or approximately $\mathbf{2.23\%}$.

Given:

Service Date: 27-05-2009.

(a) Engine oil quantity: $3.10$ litres.

Engine oil rate: $\textsf{₹}$ $178.75$ per litre.

VAT on engine oil: $20\%$.

(b) Charges for other services: $\textsf{₹}$ $1,105.12$.

VAT on other services: $12.5\%$.

(c) Labour charges: $\textsf{₹}$ $2,095.80$.

Service tax on labour charges: $10\%$.

(d) Cess on service tax: $3\%$.

To Find:

The total bill amount.

Solution:

Let's calculate the cost for each component of the bill:

(a) Engine Oil Cost:

Cost of engine oil before VAT = Quantity $\times$ Rate

Cost of engine oil before VAT = $3.10 \times 178.75$

$\begin{array}{cc}& & & 1 & 7 & 8 & . & 7 & 5 \\ \times & & & & & 3 & . & 1 & 0 \\ \hline &&&&& 0 & 0 & 0 & 0 & 0 \\ &&& 1 & 7 & 8 & 7 & 5 & \times \\ && 5 & 3 & 6 & 2 & 5 & \times & \times \\ \hline && 5 & 5 & 4 & . & 1 & 2 & 5 & 0 \\ \hline \end{array}$

Cost of engine oil before VAT = $\textsf{₹}$ $554.125$.

VAT on engine oil = $20\%$ of $\textsf{₹}$ $554.125$

VAT on engine oil = $0.20 \times 554.125 = \textsf{₹} \ 110.825$.

Total cost for engine oil (including VAT) = $554.125 + 110.825 = \textsf{₹} \ 664.95$.

(b) Other Services Cost:

Cost of other services before VAT = $\textsf{₹}$ $1,105.12$.

VAT on other services = $12.5\%$ of $\textsf{₹}$ $1,105.12$

VAT on other services = $0.125 \times 1105.12$

$\begin{array}{cc}& & 1 & 1 & 0 & 5 & . & 1 & 2 \\ \times & & & 0 & . & 1 & 2 & 5 \\ \hline &&& 5 & 5 & 2 & 5 & 6 & 0 \\ && 2 & 2 & 1 & 0 & 2 & 4 & \times \\ & 1 & 1 & 0 & 5 & 1 & 2 & \times & \times \\ \hline & 1 & 3 & 8 & . & 1 & 4 & 0 & 0 & 0 \\ \hline \end{array}$

VAT on other services = $\textsf{₹}$ $138.14$ (rounded to two decimal places).

Total cost for other services (including VAT) = $1105.12 + 138.14 = \textsf{₹} \ 1243.26$.

(c) and (d) Labour Charges and Taxes:

Labour charges = $\textsf{₹}$ $2,095.80$.

Service tax = $10\%$ of $\textsf{₹}$ $2,095.80$

Service tax = $0.10 \times 2095.80 = \textsf{₹} \ 209.58$.

Cess on service tax = $3\%$ of Service tax

Cess = $3\%$ of $\textsf{₹}$ $209.58$

Cess = $0.03 \times 209.58$

$\begin{array}{cc}& & & 2 & 0 & 9 & . & 5 & 8 \\ \times & & & & & 0 & . & 0 & 3 \\ \hline &&& 6 & . & 2 & 8 & 7 & 4 \\ \hline \end{array}$

Cess = $\textsf{₹}$ $6.2874$ (rounded to $\textsf{₹}$ $6.29$).

Total cost for labour (including taxes) = Labour charges + Service tax + Cess

Total cost for labour = $2095.80 + 209.58 + 6.29 = \textsf{₹} \ 2311.67$.

Total Bill Amount:

Total Bill = Total Engine Oil Cost + Total Other Services Cost + Total Labour Cost

Total Bill = $664.95 + 1243.26 + 2311.67$

$\begin{array}{ccccccc} & & 6 & 6 & 4 & . 9 & 5 \\ & & 1 & 2 & 4 & 3 & . 2 & 6 \\ + & & 2 & 3 & 1 & 1 & . 6 & 7 \\ \hline & & 4 & 2 & 1 & 9 & . 8 & 8 \\ \hline \end{array}$

Total Bill Amount = $\textsf{₹}$ $4219.88$.

Final Answer:

The total bill amount is $\mathbf{\textsf{₹} \ 4219.88}$.

Given:

Principal ($P$) = $\textsf{₹}$ $40,000$.

Rate of Interest ($R$) = $8\%$ p.a. compounded annually.

To Find:

(a) Interest for the first year.

(b) Principal for the second year.

(c) Interest for the second year.

(d) Amount after two years.

Solution:

(a) Interest if period is one year:

Since the interest is compounded annually, the interest for the first year is the same as simple interest on the original principal.

Interest for 1st year ($I_1$) = $P \times R \times \text{Time}$

$I_1 = 40000 \times \frac{8}{100} \times 1$

$I_1 = 40000 \times 0.08$

$I_1 = 3200$

The interest for the first year is $\textsf{₹}$ $3200$.

(b) Principal for 2nd year:

The principal for the second year is the amount at the end of the first year.

Amount at the end of 1st year ($A_1$) = Principal + Interest for 1st year

$A_1 = P + I_1$

$A_1 = 40000 + 3200$

$A_1 = 43200$

The principal for the 2nd year is $\textsf{₹}$ $43200$.

(c) Interest for 2nd year:

The interest for the second year is calculated on the principal of the second year.

Interest for 2nd year ($I_2$) = Principal for 2nd year $\times$ Rate $\times$ Time

$I_2 = 43200 \times \frac{8}{100} \times 1$

$I_2 = 43200 \times 0.08$

$I_2 = 3456$

The interest for the 2nd year is $\textsf{₹}$ $3456$.

(d) Amount if period is 2 years:

The amount after 2 years is the sum of the principal for the 2nd year and the interest for the 2nd year.

Amount at the end of 2nd year ($A_2$) = Principal for 2nd year + Interest for 2nd year

$A_2 = 43200 + 3456$

$A_2 = 46656$

Alternatively, using the compound interest formula $A = P(1 + R)^n$:

$A_2 = 40000 \left(1 + \frac{8}{100}\right)^2$

$A_2 = 40000 (1 + 0.08)^2$

$A_2 = 40000 (1.08)^2$

$A_2 = 40000 \times 1.1664$

$A_2 = 46656$

The amount after 2 years is $\textsf{₹}$ $46656$.

Final Answer:

(a) Interest if period is one year: $\mathbf{\textsf{₹} \ 3200}$.

(b) Principal for 2nd year: $\mathbf{\textsf{₹} \ 43200}$.

(c) Interest for 2nd year: $\mathbf{\textsf{₹} \ 3456}$.

(d) Amount if period is 2 years: $\mathbf{\textsf{₹} \ 46656}$.

Given:

Total number of seats available = $49,000$.

Number of seats for General Category = $28,200$.

Number of seats for SC Category = $7,400$.

Number of seats for ST Category = $3,700$.

To Find:

(i) The percentage of seats available for Students of General Category.

(ii) The percentage of seats available for Students of SC Category and ST Category taken together.

Solution:

(i) Percentage of seats for General Category:

Percentage = $\frac{\text{Number of General Category Seats}}{\text{Total Number of Seats}} \times 100\%$

Percentage = $\frac{28200}{49000} \times 100\%$

Percentage = $\frac{282}{490} \times 100\%$

Percentage = $\frac{282}{49} \times 10\%$

Percentage = $\frac{2820}{49}\%$

Dividing $2820$ by $49$:

$2820 \div 49 \approx 57.55$

The percentage of seats for General Category is $\frac{2820}{49}\%$ or approximately $57.55\%$.

(ii) Percentage of seats for SC Category and ST Category taken together:

Total number of seats for SC and ST Categories = Number of SC Seats + Number of ST Seats

Total reserved seats (SC + ST) = $7400 + 3700 = 11100$

Percentage = $\frac{\text{Total Number of SC and ST Seats}}{\text{Total Number of Seats}} \times 100\%$

Percentage = $\frac{11100}{49000} \times 100\%$

Percentage = $\frac{111}{490} \times 100\%$

Percentage = $\frac{111}{49} \times 10\%$

Percentage = $\frac{1110}{49}\%$

Dividing $1110$ by $49$:

$1110 \div 49 \approx 22.65$

The percentage of seats for SC and ST Categories taken together is $\frac{1110}{49}\%$ or approximately $22.65\%$.

Verification:

Let's find the number of seats for Other Backward Classes (OBC) and their percentage, assuming the remaining seats are for OBC.

OBC Seats = Total Seats - General Seats - SC Seats - ST Seats

OBC Seats = $49000 - 28200 - 7400 - 3700 = 49000 - 39300 = 9700$.

Percentage of OBC Seats = $\frac{9700}{49000} \times 100\% = \frac{97}{490} \times 100\% = \frac{970}{49}\% \approx 19.80\%$.

Total Percentage $\approx 57.55\% + 22.65\% + 19.80\% = 100\%$. (Sum of fractional values would be exactly 100%)

Final Answer:

(i) The percentage of seats available for Students of General Category is $\mathbf{\frac{2820}{49}\%}$ or approximately $\mathbf{57.55\%}$.

(ii) The percentage of seats available for Students of SC Category and ST Category taken together is $\mathbf{\frac{1110}{49}\%}$ or approximately $\mathbf{22.65\%}$.

Given:

Total price of the medicines (including VAT) = $\textsf{₹}$ $36.40$.

VAT percentage = $4\%$.

To Find:

The price of the medicines before VAT was added.

Solution:

Let the price of the medicines before VAT be $P$.

The sales tax (VAT) is calculated on the original price $P$.

VAT Amount = VAT Percentage $\times$ Original Price

VAT Amount = $4\%$ of $P$

VAT Amount = $\frac{4}{100} \times P = 0.04 P$

The total price is the sum of the original price and the VAT amount.

Total Price = Original Price + VAT Amount

$36.40 = P + 0.04 P$

Combine the terms with $P$:

$36.40 = (1 + 0.04) P$

$36.40 = 1.04 P$

To find $P$, divide the total price by $1.04$:

$P = \frac{36.40}{1.04}$

To remove the decimal, multiply the numerator and denominator by $100$:

$P = \frac{36.40 \times 100}{1.04 \times 100}$

$P = \frac{3640}{104}$

Simplify the fraction. Both numerator and denominator are divisible by $4$:

$P = \frac{\cancel{3640}^{910}}{\cancel{104}_{26}}$

Both numerator and denominator are divisible by $2$:

$P = \frac{\cancel{910}^{455}}{\cancel{26}_{13}}$

Now, divide $455$ by $13$:

$455 \div 13 = 35$

$P = 35$

The price before VAT was added is $\textsf{₹}$ $35$.

Verification:

Price before VAT = $\textsf{₹}$ $35$.

VAT amount = $4\%$ of $\textsf{₹}$ $35 = \frac{4}{100} \times 35 = 0.04 \times 35 = \textsf{₹} \ 1.40$.

Total price = Price before VAT + VAT amount = $35 + 1.40 = \textsf{₹} \ 36.40$.

This matches the given total price.

Final Answer:

The price before VAT was added is $\mathbf{\textsf{₹} \ 35}$.

Given:

Total amount paid (including taxes) = $\textsf{₹}$ $387$.

Amount of taxes included = $\textsf{₹}$ $43$.

To Find:

The tax percentage.

Solution:

First, we need to find the price of the items before the tax was added.

Price before Tax = Total Amount Paid - Amount of Taxes

Price before Tax = $387 - 43$

Price before Tax = $\textsf{₹}$ $344$.

The tax percentage is calculated on the price before tax.

Tax Percentage = $\frac{\text{Amount of Taxes}}{\text{Price before Tax}} \times 100\%$

Tax Percentage = $\frac{43}{344} \times 100\%$

Simplify the fraction $\frac{43}{344}$. We notice that $344 = 8 \times 43$.

Tax Percentage = $\frac{\cancel{43}^{1}}{\cancel{344}_{8}} \times 100\%$

Tax Percentage = $\frac{1}{8} \times 100\%$

Tax Percentage = $\frac{100}{8}\%$

Simplify the fraction $\frac{100}{8}$. Both are divisible by $4$.

Tax Percentage = $\frac{\cancel{100}^{25}}{\cancel{8}_{2}}\%$

Tax Percentage = $\frac{25}{2}\% = 12.5\%$.

The tax percentage is $12.5\%$.

Final Answer:

The tax percentage is $\mathbf{12.5\%}$.

Given:

Details of household items purchased, their Marked Price (Rate/Amount), and the discount percentage offered are provided in the table.

To Find:

The total amount of the bill to be paid after applying discounts.

Solution:

We need to calculate the Selling Price (S.P.) for each item after the discount and then sum them up to find the total bill amount.

S.P. = M.P. - Discount Amount

Discount Amount = M.P. $\times$ Discount Percentage

So, S.P. = M.P. $\times$ $(1 - \text{Discount Rate})$

(a) Atta:

M.P. = $\textsf{₹}$ $200$

Discount % = $16\%$

Discount Amount = $16\%$ of $200 = \frac{16}{100} \times 200 = 16 \times 2 = \textsf{₹} \ 32$.

S.P. (Atta) = $200 - 32 = \textsf{₹} \ 168$.

(b) Detergent:

M.P. = $\textsf{₹}$ $371$

Discount % = $22.10\%$

Discount Amount = $22.10\%$ of $371 = \frac{22.10}{100} \times 371 = 0.221 \times 371$.

$\begin{array}{cc}& & & 3 & 7 & 1 \\ \times & & & 0 & . 2 & 2 & 1 \\ \hline &&& & 3 & 7 & 1 \\ &&& 7 & 4 & 2 & \times \\ && 7 & 4 & 2 & \times & \times \\ \hline && 8 & 2 & . 0 & 4 & 1 \\ \hline \end{array}$

Discount Amount $\approx \textsf{₹} \ 82.04$ (rounding to two decimal places).

S.P. (Detergent) = $371 - 82.04 = \textsf{₹} \ 288.96$.

(c) Namkeen:

M.P. = $\textsf{₹}$ $153$

Discount % = $18.30\%$

Discount Amount = $18.30\%$ of $153 = \frac{18.30}{100} \times 153 = 0.183 \times 153$.

$\begin{array}{cc}& & & 1 & 5 & 3 \\ \times & & & 0 & . 1 & 8 & 3 \\ \hline &&& & 4 & 5 & 9 \\ && 1 & 2 & 2 & 4 & \times \\ && 1 & 5 & 3 & \times & \times \\ \hline && 2 & 8 & . 0 & 0 & 0 & 9 \\ \hline \end{array}$

Discount Amount $\approx \textsf{₹} \ 28.01$ (rounding to two decimal places).

S.P. (Namkeen) = $153 - 28.01 = \textsf{₹} \ 124.99$.

Total Bill Amount:

Total Bill = S.P. (Atta) + S.P. (Detergent) + S.P. (Namkeen)

Total Bill = $168 + 288.96 + 124.99$.

$\begin{array}{ccccccc} & & 1 & 6 & 8 & . 0 & 0 \\ & & 2 & 8 & 8 & . 9 & 6 \\ + & & 1 & 2 & 4 & . 9 & 9 \\ \hline & & 5 & 8 & 1 & . 9 & 5 \\ \hline \end{array}$

Total Bill Amount = $\textsf{₹} \ 581.95$.

Final Answer:

The total amount of the bill she has to pay is $\mathbf{\textsf{₹} \ 581.95}$.

Given:

Subscription charges for period 1 (17-02-09 to 23-02-09) = $\textsf{₹}$ $199.75$.

Service tax for period 1 = $12\%$.

Subscription charges for period 2 (24-02-09 to 16-03-09) = $\textsf{₹}$ $599.25$.

Service tax for period 2 = $10\%$.

Education cess on service tax = $3\%$.

To Find:

The final bill amount.

Solution:

First, calculate the service tax amount for each period.

Service tax for Period 1 = $12\%$ of $\textsf{₹}$ $199.75$

Service tax for Period 1 = $\frac{12}{100} \times 199.75 = 0.12 \times 199.75 = \textsf{₹} \ 23.97$.

Service tax for Period 2 = $10\%$ of $\textsf{₹}$ $599.25$

Service tax for Period 2 = $\frac{10}{100} \times 599.25 = 0.10 \times 599.25 = \textsf{₹} \ 59.925$.

Next, calculate the total service tax amount.

Total Service Tax = Service tax for Period 1 + Service tax for Period 2

Total Service Tax = $23.97 + 59.925 = \textsf{₹} \ 83.895$.

Now, calculate the education cess amount on the total service tax.

Education Cess = $3\%$ of Total Service Tax

Education Cess = $3\%$ of $\textsf{₹}$ $83.895$

Education Cess = $\frac{3}{100} \times 83.895 = 0.03 \times 83.895 = \textsf{₹} \ 2.51685$.

The base bill amount is the sum of the charges for both periods before tax.

Base Bill Amount = Charges for Period 1 + Charges for Period 2

Base Bill Amount = $199.75 + 599.25 = \textsf{₹} \ 799.00$.

Finally, calculate the total bill amount by adding the base amount, total service tax, and education cess.

Final Bill Amount = Base Bill Amount + Total Service Tax + Education Cess

Final Bill Amount = $799.00 + 83.895 + 2.51685$

Final Bill Amount = $885.41185$

Rounding the final amount to two decimal places:

Final Bill Amount $\approx \textsf{₹} \ 885.41$.

Final Answer:

The final bill amount is $\mathbf{\textsf{₹} \ 885.41}$ (rounded to two decimal places).

Given:

Principal ($P$) = $\textsf{₹}$ $1,00,000$.

Rate of Interest (Annual) = $10\%$ p.a.

Compounding Frequency: Half yearly.

Derived Rate and Time Period:

Since interest is compounded half-yearly, the half-yearly rate is half of the annual rate.

Half-yearly Rate ($r$) = $\frac{10\%}{2} = 5\% = \frac{5}{100} = 0.05$.

Each period is 6 months (a half-year).

To Find:

(i) Interest for the first 6 months.

(ii) Amount after 6 months.

(iii) Interest for the next 6 months (second half-year).

(iv) Amount after one year (two half-years).

Solution:

(i) Interest for 6 months:

This is the interest for the first compounding period (1 half-year).

Interest = Principal $\times$ Rate per period $\times$ Number of periods

Interest for first 6 months ($I_1$) = $P \times r \times 1$

$I_1 = 100000 \times 0.05 \times 1$

$I_1 = 5000$

The interest for the first 6 months is $\textsf{₹}$ $5000$.

(ii) Amount after 6 months:

The amount after the first compounding period becomes the principal for the next period.

Amount after 6 months ($A_1$) = Principal + Interest for first 6 months

$A_1 = P + I_1$

$A_1 = 100000 + 5000$

$A_1 = 105000$

The amount after 6 months is $\textsf{₹}$ $105000$.

(iii) Interest for next 6 months:

This is the interest for the second compounding period (the next half-year).

The principal for this period is the amount at the end of the first period ($A_1$).

Principal for 2nd 6 months = $\textsf{₹}$ $105000$.

Interest for next 6 months ($I_2$) = Principal for 2nd 6 months $\times$ Rate per period $\times$ Number of periods

$I_2 = 105000 \times 0.05 \times 1$

$I_2 = 5250$

The interest for the next 6 months is $\textsf{₹}$ $5250$.

(iv) Amount after one year:

One year consists of two half-yearly compounding periods.

The amount after one year is the amount at the end of the second half-year.

Amount after one year ($A_2$) = Amount after 6 months + Interest for next 6 months

$A_2 = A_1 + I_2$

$A_2 = 105000 + 5250$

$A_2 = 110250$

Alternatively, using the compound interest formula for $n=2$ half-yearly periods:

$A = P(1 + r)^n$

$A_2 = 100000(1 + 0.05)^2$

$A_2 = 100000(1.05)^2$

$A_2 = 100000 \times 1.1025$

$A_2 = 110250$

The amount after one year is $\textsf{₹}$ $110250$.

Final Answer:

(i) Interest for 6 months: $\mathbf{\textsf{₹} \ 5000}$.

(ii) Amount after 6 months: $\mathbf{\textsf{₹} \ 105000}$.

(iii) Interest for next 6 months: $\mathbf{\textsf{₹} \ 5250}$.

(iv) Amount after one year: $\mathbf{\textsf{₹} \ 110250}$.

Given:

Total quantity of mangoes bought = $160$ kg.

Cost Price (C.P.) per kg = $\textsf{₹}$ $48$.

$70\%$ of mangoes sold at $\textsf{₹}$ $70$ per kg.

Remaining mangoes sold at $\textsf{₹}$ $40$ per kg.

To Find:

The overall gain or loss percentage.

Solution:

First, calculate the total Cost Price (C.P.) of the mangoes.

Total C.P. = Quantity $\times$ C.P. per kg

Total C.P. = $160 \times 48$

$\begin{array}{cc} & 1 & 6 & 0 \\ \times & & 4 & 8 \\ \hline & 1 & 2 & 8 & 0 \\ 6 & 4 & 0 & \times \\ \hline 7 & 6 & 8 & 0 \\ \hline \end{array}$

Total C.P. = $\textsf{₹}$ $7680$.

Next, calculate the quantities sold at different rates.

Quantity sold at $\textsf{₹}$ $70$/kg = $70\%$ of $160$ kg

Quantity 1 = $\frac{70}{100} \times 160 = 0.70 \times 160 = 112$ kg.

Remaining quantity sold at $\textsf{₹}$ $40$/kg = Total Quantity - Quantity 1

Quantity 2 = $160 - 112 = 48$ kg.

Now, calculate the Selling Price (S.P.) for each part.

S.P. of Quantity 1 = Quantity 1 $\times$ S.P. per kg for Quantity 1

S.P. 1 = $112 \times 70$

$\begin{array}{cc}& & 1 & 1 & 2 \\ \times & & & 7 & 0 \\ \hline &&& 0 & 0 & 0 \\ & 7 & 8 & 4 & \times \\ \hline & 7 & 8 & 4 & 0 \\ \hline \end{array}$

S.P. 1 = $\textsf{₹}$ $7840$.

S.P. of Quantity 2 = Quantity 2 $\times$ S.P. per kg for Quantity 2

S.P. 2 = $48 \times 40 = 1920$.

S.P. 2 = $\textsf{₹}$ $1920$.

Calculate the total Selling Price (S.P.).

Total S.P. = S.P. 1 + S.P. 2

Total S.P. = $7840 + 1920 = \textsf{₹} \ 9760$.

Compare Total S.P. and Total C.P. to determine gain or loss.

Total S.P. ($\textsf{₹} \ 9760$) > Total C.P. ($\textsf{₹} \ 7680$).

Since Total S.P. is greater than Total C.P., there is a gain.

Gain Amount = Total S.P. - Total C.P.

Gain Amount = $9760 - 7680$

$\begin{array}{cc} & 9 & 7 & 6 & 0 \\ - & 7 & 6 & 8 & 0 \\ \hline & 2 & 0 & 8 & 0 \\ \hline \end{array}$

Gain Amount = $\textsf{₹}$ $2080$.

Calculate the gain percentage.

Gain Percentage = $\frac{\text{Gain Amount}}{\text{Total C.P.}} \times 100\%$

Gain Percentage = $\frac{2080}{7680} \times 100\%$

Simplify the fraction:

Gain Percentage = $\frac{208}{768} \times 100\%$

Gain Percentage = $\frac{\cancel{208}^{13}}{\cancel{768}_{48}} \times 100\%$ (Dividing numerator and denominator by 16)

Gain Percentage = $\frac{13}{48} \times 100\%$

Gain Percentage = $\frac{1300}{48}\%$

Simplify by dividing numerator and denominator by 4:

Gain Percentage = $\frac{\cancel{1300}^{325}}{\cancel{48}_{12}}\%$

Gain Percentage = $\frac{325}{12}\%$.

Convert to a mixed number:

$\frac{325}{12} = \frac{12 \times 27 + 1}{12} = 27\frac{1}{12}\%$.

As a decimal: $325 \div 12 \approx 27.0833...$

Final Answer:

Babita made a gain. The gain percentage on the whole dealing is $\mathbf{27\frac{1}{12}\%}$ or approximately $\mathbf{27.08\%}$.

Let the marked price of the skirt be MP.

Given:

Marked Price (MP) = $\textsf{₹} 1200$

Initial Discount = $25\%$

Additional Off-Season Discount = $30\%$ on the discounted price.

To Find:

The final price Pragya paid for the skirt.

Solution:

First, calculate the price after the initial $25\%$ discount.

Discount amount = $25\%$ of MP

Price after initial discount = MP - Discount amount

Now, an additional $30\%$ discount is offered on this discounted price ($\textsf{₹} 900$).

Additional discount amount = $30\%$ of the price after initial discount

Final price = Price after initial discount - Additional discount amount

Alternatively, we can calculate the price remaining after each discount.

After a $25\%$ discount, the customer pays $100\% - 25\% = 75\%$ of the price.

Price after initial discount = $75\%$ of $1200 = 0.75 \times 1200 = \textsf{₹} 900$.

After an additional $30\%$ discount on $\textsf{₹} 900$, the customer pays $100\% - 30\% = 70\%$ of the $\textsf{₹} 900$.

Final price = $70\%$ of $900 = 0.70 \times 900 = \textsf{₹} 630$.

Answer:

Pragya paid $\textsf{₹} 630$ for the skirt.

Let the marked price of the mobile phone be MP.

Given:

Discount Percentage = $25\%$

Discount Amount = $\textsf{₹} 1960$

To Find:

The Marked Price (MP) of the mobile phone.

Solution:

The discount amount is calculated as a percentage of the marked price.

Discount Amount = Discount Percentage $\times$ Marked Price

We are given the discount amount and the discount percentage, and we need to find the marked price (MP).

So, we can write the equation:

Convert the percentage to a decimal or fraction:

Simplify the fraction $\frac{25}{100}$ to $\frac{1}{4}$.

To find MP, multiply both sides of the equation by 4:

Perform the multiplication:

Thus, the marked price of the mobile phone was $\textsf{₹} 7840$.

Verification:

Calculate $25\%$ of $\textsf{₹} 7840$:

$25\%$ of $\textsf{₹} 7840 = 0.25 \times 7840 = \frac{1}{4} \times 7840 = \textsf{₹} 1960$.

This matches the given discount amount.

Answer:

The marked price of the mobile phone was $\textsf{₹} 7840$.

Let P be the principal amount, R be the rate of interest per annum, and T be the time period in years.

Given:

Principal (P) = $\textsf{₹} 45000$

Rate (R) = $12\%$ per annum

Time (T) = $5$ years

To Find:

The difference between Compound Interest (CI) and Simple Interest (SI).

Solution:

First, calculate the Simple Interest (SI).

The formula for Simple Interest is:

$SI = \frac{P \times R \times T}{100}$

Substitute the given values:

Cancel out common factors:

So, the Simple Interest is $\textsf{₹} 27000$.

Next, calculate the Compound Interest (CI).

The formula for the amount (A) with Compound Interest is:

$A = P \left(1 + \frac{R}{100}\right)^T$

Substitute the given values:

Now, calculate $(1.12)^5$:

$(1.12)^2 = 1.2544$

$(1.12)^3 = 1.2544 \times 1.12 = 1.404928$

$(1.12)^4 = 1.404928 \times 1.12 = 1.57351936$

$(1.12)^5 = 1.57351936 \times 1.12 = 1.7623416832$

Substitute this value back into the formula for A:

Round the amount to two decimal places for currency:

The Compound Interest (CI) is the difference between the Amount and the Principal:

$CI = A - P$

So, the Compound Interest is $\textsf{₹} 34305.38$.

Finally, find the difference between Compound Interest and Simple Interest.

Difference = CI - SI

The difference is $\textsf{₹} 7305.38$.

Answer:

The difference between the Compound Interest and Simple Interest is $\textsf{₹} 7305.38$.

Let the initial cost of the computer be $C_0$.

Given:

Initial Cost ($C_0$) = $\textsf{₹} 1,00,000$

Annual Depreciation Rate (r) = $50\%$

Time period (n) = $2$ years

To Find:

The cost of the computer after two years ($C_2$).

Solution:

Depreciation is calculated on the value of the asset at the beginning of the year.

After Year 1:

Depreciation in Year 1 = $50\%$ of Initial Cost

Cost after Year 1 ($C_1$) = Initial Cost - Depreciation in Year 1

After Year 2:

The depreciation in Year 2 is calculated on the value at the beginning of Year 2, which is $C_1$.

Depreciation in Year 2 = $50\%$ of Cost after Year 1

Cost after Year 2 ($C_2$) = Cost after Year 1 - Depreciation in Year 2

The cost of the computer after two years will be $\textsf{₹} 25,000$.

Alternate Solution:

We can use the formula for depreciation: $C_n = C_0 (1 - r)^n$, where $C_n$ is the cost after n years, $C_0$ is the initial cost, r is the annual depreciation rate (as a decimal), and n is the number of years.

Given $C_0 = \textsf{₹} 1,00,000$, $r = 50\% = 0.50$, and $n = 2$ years.

This confirms the result obtained from the step-by-step method.

Answer:

The cost of the computer after two years will be $\textsf{₹} 25,000$.

Let the population of the town two years ago be $P_0$.

Given:

Present population ($P_2$) = $6,31,680$

Decrease rate last year (Year 2) = $4\%$

Decrease rate the year before last (Year 1) = $6\%$

To Find:

The population two years ago ($P_0$).

Solution:

The population decreases each year based on the population at the beginning of that year.

Let $P_1$ be the population last year (which is the population after the first year's decrease).

The population decreased by $6\%$ in the first year (two years ago to last year).

So, the population last year ($P_1$) is $P_0$ reduced by $6\%$.

In the second year (last year to present year), the population decreased by $4\%$ based on the population at the beginning of that year, which is $P_1$.

So, the present population ($P_2$) is $P_1$ reduced by $4\%$.

Now, substitute the expression for $P_1$ from equation (i) into equation (ii):

We are given that the present population $P_2 = 6,31,680$. Substitute this value into equation (iii):

Calculate the product $0.94 \times 0.96$:

So, the equation becomes:

To find $P_0$, divide $631680$ by $0.9024$:

To perform the division, we can write $0.9024$ as a fraction $\frac{9024}{10000}$.

We can perform the division $\frac{631680}{9024}$:

Substitute this back into the expression for $P_0$:

So, the population two years ago was $7,00,000$.

Verification:

Population two years ago = $700000$.

After 6% decrease (Year 1): $700000 \times (1 - 0.06) = 700000 \times 0.94 = 658000$.

Population last year = $658000$.

After 4% decrease (Year 2): $658000 \times (1 - 0.04) = 658000 \times 0.96$.

This matches the given present population, $6,31,680$.

Answer:

The population two years ago was $7,00,000$.

Let's determine the cost price and selling price per lemon to find the gain or loss percentage.

Given:

Cost Price of 1 dozen (12) lemons = $\textsf{₹} 48$

Selling Price of 10 lemons = $\textsf{₹} 40$

To Find:

The gain or loss per cent.

Solution:

First, calculate the Cost Price (CP) of one lemon.

CP of 1 dozen lemons = $\textsf{₹} 48$

Since 1 dozen = 12 lemons, the CP of 12 lemons is $\textsf{₹} 48$.

Calculate the value:

Next, calculate the Selling Price (SP) of one lemon.

SP of 10 lemons = $\textsf{₹} 40$

Calculate the value:

Now, compare the CP and SP per lemon.

From (i), CP of 1 lemon = $\textsf{₹} 4$.

From (ii), SP of 1 lemon = $\textsf{₹} 4$.

Since the Selling Price is equal to the Cost Price, there is neither a gain nor a loss on selling the lemons.

Gain or Loss = SP - CP

The gain or loss percentage is calculated as:

Gain/Loss $\% = \frac{\text{Gain or Loss}}{\text{CP}} \times 100\%$

Answer:

There is neither gain nor loss. The gain or loss per cent is $0\%$.

We need to calculate the percentage decrease in price for each fuel type. The formula for percentage decrease is:

Percentage Decrease $= \frac{\text{Old Price} - \text{New Price}}{\text{Old Price}} \times 100\%$

Calculation for Petrol:

Old Price = $\textsf{₹} 45.62$

New Price = $\textsf{₹} 40.62$

Decrease Amount = Old Price - New Price = $45.62 - 40.62 = \textsf{₹} 5.00$

Calculation for Diesel:

Old Price = $\textsf{₹} 32.86$

New Price = $\textsf{₹} 30.86$

Decrease Amount = Old Price - New Price = $32.86 - 30.86 = \textsf{₹} 2.00$

Calculation for LPG:

Old Price = $\textsf{₹} 304.70$

New Price = $\textsf{₹} 279.70$

Decrease Amount = Old Price - New Price = $304.70 - 279.70 = \textsf{₹} 25.00$

Completed Table:

We need to calculate the percentage change in the number of seats for each political party from 2004 to 2009. The formula for percentage change (increase or decrease) is:

Percentage Change $= \frac{\text{Number of seats in 2009} - \text{Number of seats in 2004}}{\text{Number of seats in 2004}} \times 100\%$

If the result is positive, it is a percentage increase; if negative, it is a percentage decrease.

Political Party A:

Seats in 2004 = $206$

Seats in 2009 = $145$

Change in seats = $145 - 206 = -61$

Since the change is negative, it is a decrease.

Result for Party A: $29.61\%$ Decrease

Political Party B:

Seats in 2004 = $116$

Seats in 2009 = $138$

Change in seats = $138 - 116 = +22$

Since the change is positive, it is an increase.

Result for Party B: $18.97\%$ Increase

Political Party C:

Seats in 2004 = $4$

Seats in 2009 = $24$

Change in seats = $24 - 4 = +20$

Since the change is positive, it is an increase.

Result for Party C: $500\%$ Increase

Political Party D:

Seats in 2004 = $11$

Seats in 2009 = $12$

Change in seats = $12 - 11 = +1$

Since the change is positive, it is an increase.

Result for Party D: $9.09\%$ Increase

Summary of Results:

Political Party A: $29.61\%$ Decrease

Political Party B: $18.97\%$ Increase

Political Party C: $500\%$ Increase

Political Party D: $9.09\%$ Increase

We need to find the percentage by which the number of seats won by Party X is more than the number of seats won by Party Y. This is a percentage increase calculation relative to the number of seats won by Party Y.

Given:

Number of seats won by Party X = $158$

Number of seats won by Party Y = $105$

To Find:

The percentage by which seats won by X are more than Y.

Solution:

First, calculate the difference in the number of seats won by X and Y.

Difference in seats = (Seats won by X) - (Seats won by Y)

This means Party X won $53$ more seats than Party Y.

To find out how much more this is in percentage compared to Y, we use the number of seats won by Y as the base.

Percentage more = $\frac{\text{Difference}}{\text{Seats won by Y}} \times 100\%$

Substitute the values:

Calculate the value:

Therefore, Party X won approximately $50.48\%$ more seats than Party Y.

Answer:

Party X won approximately $50.48\%$ more seats as compared to Party Y.

Let the Selling Price of each cooler be SP.

Given:

Selling Price of the first cooler = $\textsf{₹} 3990$

Selling Price of the second cooler = $\textsf{₹} 3990$

Gain on selling the first cooler = $5\%$

Loss on selling the second cooler = $5\%$

To Find:

The overall gain or loss percentage in the whole transaction.

Solution:

First, calculate the Cost Price (CP) of each cooler.

For the first cooler (sold at 5% gain):

Let CP$_1$ be the cost price of the first cooler.

Selling Price = Cost Price + Gain

SP$_1$ = CP$_1$ + 5% of CP$_1$

The cost price of the first cooler is $\textsf{₹} 3800$.

For the second cooler (sold at 5% loss):

Let CP$_2$ be the cost price of the second cooler.

Selling Price = Cost Price - Loss

SP$_2$ = CP$_2$ - 5% of CP$_2$

The cost price of the second cooler is $\textsf{₹} 4200$.

Now, calculate the total Selling Price and total Cost Price for the whole transaction.

Total Selling Price (Total SP) = SP$_1$ + SP$_2$

Total Cost Price (Total CP) = CP$_1$ + CP$_2$

Compare Total SP and Total CP:

Since Total SP is less than Total CP, there is an overall loss in the transaction.

Overall Loss Amount = Total CP - Total SP

Now, calculate the overall loss percentage.

Overall Loss Percentage = $\frac{\text{Overall Loss}}{\text{Total CP}} \times 100\%$

Alternate Solution (Formula Method):

When two articles are sold at the same Selling Price, and one is sold at a gain of x% while the other is sold at a loss of x%, there is always an overall loss.

The overall loss percentage is given by the formula:

Here, $x = 5$ (the common gain/loss percentage).

Substitute $x=5$ into the formula:

Both methods yield the same result.

Answer:

There is an overall loss of $0.25\%$ in the whole transaction.

Interpretation of the problem:

It is assumed that the phrase "buys some pencils for Rs 3" means the total cost of a certain number of pencils was Rs 3. Similarly, "an equal number for Rs 6" means the total cost of the same number of pencils was Rs 6. "She sells them for Rs 7" means the total selling price for all the pencils was Rs 7.

Given:

Cost of the first lot of pencils = $\textsf{₹} 3$

Cost of the second lot of pencils = $\textsf{₹} 6$

Number of pencils in the first lot = Number of pencils in the second lot

Selling price of all the pencils = $\textsf{₹} 7$

To Find:

The gain or loss percentage.

Solution:

Let the number of pencils in the first lot be $N$.

Since the number of pencils in the second lot is equal to the number in the first lot, the number of pencils in the second lot is also $N$.

The total number of pencils bought is the sum of the numbers in the two lots.

The total Cost Price (CP) is the sum of the costs of the two lots.

The total Selling Price (SP) for all the pencils is given as $\textsf{₹} 7$.

Compare Total SP and Total CP:

Since the Total Selling Price is less than the Total Cost Price, there is a loss in the transaction.

Loss Amount = Total CP - Total SP

Now, calculate the loss percentage. The loss percentage is calculated on the Total Cost Price.

Loss Percentage = $\frac{\text{Loss Amount}}{\text{Total CP}} \times 100\%$

To express this as a decimal or mixed fraction:

or

Answer:

There is a loss of $\mathbf{22.\overline{2}\%}$ (or $\mathbf{22 \frac{2}{9}\%}$).

Let the Cost Price of the chair be CP.

Given:

Selling Price when there is a loss (SP$_1$) = $\textsf{₹} 736$

Loss Percentage = $8\%$

Desired Gain Percentage = $8\%$

To Find:

The Selling Price required to gain $8\%$ (SP$_2$).

Solution:

When the shopkeeper suffers a loss of $8\%$, the selling price is $100\% - 8\% = 92\%$ of the Cost Price.

We are given SP$_1 = \textsf{₹} 736$. So,

Now, we can find the Cost Price (CP).

Let's perform the division:

The Cost Price of the chair is $\textsf{₹} 800$.

Now, the shopkeeper wants to sell the chair to gain $8\%$. This means the new selling price (SP$_2$) should be $100\% + 8\% = 108\%$ of the Cost Price.

Substitute the value of CP from equation (ii):

Perform the multiplication:

So, the shopkeeper should sell the chair for $\textsf{₹} 864$ to gain $8\%$.

Answer:

He should sell it at $\textsf{₹} 864$ to gain $8\%$.

Let the Cost Price of the dining table for the shopkeeper be CP.

Given:

Cost Price (CP) for the shopkeeper = $\textsf{₹} 3200$

Shopkeeper's Gain Percentage = $6\%$

Sales Tax Rate = $5\%$ on the Selling Price.

To Find:

The total amount the customer pays for the table.

Solution:

First, calculate the Selling Price (SP) of the dining table by the shopkeeper.

The shopkeeper sells the table at a gain of $6\%$.

Selling Price = Cost Price + Gain

SP = CP + 6% of CP

The selling price of the dining table is $\textsf{₹} 3392$. This is the price before sales tax.

Now, calculate the sales tax paid by the customer. The sales tax is $5\%$ on the Selling Price.

Sales Tax Amount = $5\%$ of SP

The sales tax amount is $\textsf{₹} 169.60$.

The total amount paid by the customer is the Selling Price plus the Sales Tax Amount.

Total Amount Paid = SP + Sales Tax Amount

The customer pays a total of $\textsf{₹} 3561.60$ for the table.

Alternate Calculation for SP:

SP = CP $\times (1 + \text{Gain Percentage})$

Alternate Calculation for Total Amount Paid:

The total amount paid by the customer is the Selling Price plus $5\%$ sales tax on the selling price. This means the customer pays $100\% + 5\% = 105\%$ of the Selling Price.

Total Amount Paid = $105\%$ of SP

Both methods confirm the final amount paid by the customer.

Answer:

The customer pays a total of $\textsf{₹} 3561.60$ for the table.

To find the profit percentage, we need to calculate the total cost price and the selling price.

Given:

Original cost of the second-hand car = $\textsf{₹} 2,25,000$

Repair cost = $\textsf{₹} 25,000$

Selling Price (SP) = $\textsf{₹} 3,25,000$

To Find:

Achal's profit per cent.

Solution:

The total cost price (CP) is the sum of the original purchase price and the repair cost.

The Selling Price (SP) is given as $\textsf{₹} 3,25,000$.

Since the Selling Price is greater than the Total Cost Price, there is a profit.

Profit = Selling Price - Total Cost Price

Now, calculate the profit percentage. The profit percentage is calculated on the Total Cost Price.

Profit Percentage = $\frac{\text{Profit}}{\text{Total CP}} \times 100\%$

Substitute the values:

Simplify the fraction:

Achal's profit percentage is $30\%$.

Answer:

His profit per cent is $30\%$.

To determine the selling price for a 15% gain, we first need to calculate the total cost price incurred by the lady.

Given:

Purchase price of the air-conditioner = $\textsf{₹} 15,200$

Transportation cost = $\textsf{₹} 300$

Repair cost = $\textsf{₹} 500$

Desired Gain Percentage = $15\%$

To Find:

The Selling Price required to make a gain of $15\%$.

Solution:

The total cost price (CP) includes the purchase price and all additional expenses incurred to get the item in usable condition.

The total cost price of the air-conditioner for the lady is $\textsf{₹} 16000$.

She wants to sell it to make a gain of $15\%$. The gain is calculated on the total cost price.

Gain Amount = $15\%$ of Total CP

The required gain amount is $\textsf{₹} 2400$.

The Selling Price (SP) is the Total Cost Price plus the Gain Amount.

Alternatively, we can calculate the selling price directly:

SP = Total CP $\times (1 + \text{Gain Percentage})$

Both methods show that she should sell the air-conditioner for $\textsf{₹} 18400$ to make a gain of $15\%$.

Answer:

She should sell it at $\textsf{₹} 18400$ to make a gain of $15\%$.

Let the Cost Price of the article for the shopkeeper be CP and the Marked Price be MP.

Given:

Cost Price (CP) = $\textsf{₹} 600$

Desired Gain Percentage = $20\%$

Allowed Discount Percentage = $10\%$

To Find:

The Marked Price (MP) of the article.

Solution:

First, calculate the Selling Price (SP) at which the article should be sold to achieve a $20\%$ gain on the Cost Price.

Selling Price = Cost Price + Gain

SP = CP + 20% of CP

So, the shopkeeper must sell the article for $\textsf{₹} 720$ to get a $20\%$ profit.

Next, consider the discount offered. The discount is given on the Marked Price.

Selling Price = Marked Price - Discount

SP = MP - 10% of MP

We have two expressions for the Selling Price (SP). Equate equation (i) and equation (ii):

Now, solve for MP:

The shopkeeper should mark the article at $\textsf{₹} 800$.

Verification:

Marked Price = $\textsf{₹} 800$

Discount = $10\%$ of $\textsf{₹} 800 = 0.10 \times 800 = \textsf{₹} 80$

Selling Price = Marked Price - Discount = $800 - 80 = \textsf{₹} 720$.

Cost Price = $\textsf{₹} 600$

Profit = Selling Price - Cost Price = $720 - 600 = \textsf{₹} 120$.

Profit Percentage = $\frac{\text{Profit}}{\text{CP}} \times 100\% = \frac{120}{600} \times 100\% = \frac{1}{5} \times 100\% = 20\%$.

This matches the desired gain percentage.

Answer:

The shopkeeper should mark the article at $\textsf{₹} 800$.

Let's calculate the cost price and selling price per coat, then the total amounts.

Given:

Number of coats purchased = $18$

Cost Price per coat for Brinda = $\textsf{₹} 1500$

Brinda's Profit Percentage = $6\%$

Sales Tax Rate = $4\%$ on the Selling Price.

To Find:

1. The cost of one coat to the customer (including sales tax).

2. The total profit earned by Brinda after selling all 18 coats.

Solution:

First, calculate the Selling Price (SP) per coat for Brinda before sales tax.

Brinda sells each coat at a profit of $6\%$ on her cost price.

SP per coat = CP per coat + 6% of CP per coat

The selling price of one coat (before sales tax) is $\textsf{₹} 1590$.

Now, calculate the sales tax paid by the customer on one coat. The sales tax is $4\%$ on the Selling Price.

Sales Tax per coat = $4\%$ of SP per coat

The sales tax amount on one coat is $\textsf{₹} 63.60$.

The total cost of one coat to the customer is the Selling Price plus the Sales Tax.

Cost to Customer per coat = SP per coat + Sales Tax per coat

Next, calculate the total profit earned by Brinda.

Brinda's profit on one coat is $6\%$ of her cost price.

Profit per coat = $6\%$ of $\textsf{₹} 1500 = \textsf{₹} 90$ (as calculated in step for SP per coat).

Brinda sells 18 coats. The total profit is the profit per coat multiplied by the number of coats.

The total profit earned by Brinda is $\textsf{₹} 1620$. Note that sales tax does not affect Brinda's profit; it is an amount paid by the customer to the government (collected by the seller).

Answer:

One coat will cost $\textsf{₹} 1653.60$ to the customer.

The total profit earned by Brinda after selling all coats is $\textsf{₹} 1620$.

This problem involves calculating the amount and compound interest when the interest is compounded half-yearly.

Given:

Principal amount (P) = $\textsf{₹} 10,24,000$

Time period (T) = $1$ year

Annual Rate of Interest (R) = $5\%$ per annum

Interest is compounded half-yearly.

To Find:

1. The amount Rahim will have to pay after 1 year.

2. The interest paid by him.

Solution:

When interest is compounded half-yearly, the following adjustments are made:

New Rate per period (r) = $\frac{\text{Annual Rate}}{2} = \frac{5\%}{2} = 2.5\% = \frac{2.5}{100} = 0.025$

Number of periods (n) = Time Period in years $\times 2 = 1 \text{ year} \times 2 = 2$ half-years.

The formula for the amount (A) when interest is compounded half-yearly is:

$A = P (1 + r)^n$

Substitute the given values:

Calculate $(1.025)^2$:

Substitute this value back into the formula for A:

Perform the multiplication:

The amount Rahim will have to pay after one year is $\textsf{₹} 10,75,360$.

Next, calculate the interest paid by him.

Interest Paid = Amount - Principal

Perform the subtraction:

The interest paid by Rahim is $\textsf{₹} 51360$.

Answer:

The amount he will have to pay after the given time period is $\textsf{₹} 10,75,360$.

The interest paid by him is $\textsf{₹} 51,360$.

Let's calculate the total cost of the items before sales tax and then add the sales tax amount.

Given:

Cost of T-Shirt = $\textsf{₹} 1200$

Cost of Jeans = $\textsf{₹} 1000$

Cost of 1 Skirt = $\textsf{₹} 1350$

Number of Skirts = $2$

Sales Tax Rate = $7\%$

To Find:

The total cost of all items to Shikha, including sales tax.

Solution:

First, calculate the total cost of the items before sales tax.

Cost of 2 Skirts = $2 \times \text{Cost of 1 Skirt}$

Total Cost of Items (before tax) = Cost of T-Shirt + Cost of Jeans + Cost of 2 Skirts

The total cost of the items before sales tax is $\textsf{₹} 4900$.

Now, calculate the sales tax amount. The sales tax is $7\%$ on the total cost.

Sales Tax Amount = $7\%$ of Total Cost

The sales tax amount is $\textsf{₹} 343$.

The total cost to Shikha is the Total Cost of Items plus the Sales Tax Amount.

Alternatively, we can calculate the total cost directly. If there is a $7\%$ sales tax, the customer pays $100\% + 7\% = 107\%$ of the total cost of the items.

Cost to Shikha = $107\%$ of Total Cost

Both methods give the same total cost for Shikha.

Answer:

These items will cost $\textsf{₹} 5243$ to Shikha.

Based on the provided food labels:

Given:

• Cream of Tomato Soup: Total Fat = 2.5g, Sodium = 690mg, Sugars = 7g, Calories from Fat = 20
• Sweet Corn Soup: Total Fat = 1.5g, Sugars = 10g, Calories from Fat = 10
• Cream of Tomato Soup Sodium (690mg) is 29% of Recommended Daily Value (RDV).

To Find:

Answers to parts (a), (b), (c), and (d).

Solution:

(a) Which can be measured more accurately : the total amount of fat in cream of tomato soup or the total amount of fat in sweet corn soup? Explain.

The amount of fat in Cream of Tomato soup is $2.5$g, and in Sweet Corn soup is $1.5$g.

Both values are reported to one decimal place, implying a similar level of absolute precision in reporting (e.g., to the nearest $0.1$g or $0.05$g). However, measuring smaller quantities accurately requires more sensitive measurement techniques capable of distinguishing the amount from very low or zero values.

Since the amount of fat in Sweet Corn soup ($1.5$g) is less than the amount in Cream of Tomato soup ($2.5$g), measuring the fat content in Sweet Corn soup to a given absolute precision (like $0.1$g) represents a measurement closer to the detection limits often associated with food analysis for fat content. This implies that the measurement method used for the lower fat content might be capable of higher absolute accuracy to reliably quantify the smaller amount.

Therefore, the total amount of fat in Sweet Corn soup can be considered to be measured more accurately in terms of requiring a method capable of higher absolute precision to quantify a smaller amount.

(b) One serving of cream of tomato soup contains 29% of the recommended daily value of sodium for a 2000 calorie diet. What is the recommended daily value of sodium in milligrams? Express the answer upto 2 decimal places.

Let RDV be the Recommended Daily Value of sodium in milligrams.

From the label for Cream of Tomato soup, the sodium content is $690$mg, which is $29\%$ of the RDV.

To find the RDV, we rearrange the equation:

Now, perform the division:

Expressing the answer upto 2 decimal places:

The recommended daily value of sodium is approximately $2379.31$ mg.

(c) Find the increase per cent of sugar consumed if cream of tomato soup is chosen over sweet corn soup.

The amount of sugar in Sweet Corn soup is $10$g.

The amount of sugar in Cream of Tomato soup is $7$g.

We are asked for the "increase per cent" when choosing Cream of Tomato over Sweet Corn. This means we compare the sugar in Cream of Tomato to the sugar in Sweet Corn, using Sweet Corn as the base for comparison.

Change in sugar amount = Sugar in Cream of Tomato - Sugar in Sweet Corn

The change is negative, which means there is a decrease in the amount of sugar.

Percentage Change = $\frac{\text{Change}}{\text{Original Amount (Sweet Corn)}} \times 100\%$

Since the result is $-30\%$, there is not an increase, but a $30\%$ decrease in sugar consumed if Cream of Tomato soup is chosen over Sweet Corn soup. There is no 'increase per cent', only a decrease per cent.

(d) Calculate ratio of calories from fat in sweet corn soup to the calories from fat in cream of tomato soup.

Calories from fat in Sweet Corn soup = $10$ calories

Calories from fat in Cream of Tomato soup = $20$ calories

The ratio of calories from fat in Sweet Corn soup to Cream of Tomato soup is:

Simplify the ratio by dividing both parts by $10$:

The ratio of calories from fat in sweet corn soup to the calories from fat in cream of tomato soup is $1:2$.

Let the Original Price of the music CD be OP (which is the Marked Price).

Given:

Original Price (OP) = $\textsf{₹} 120$

Discount Percentage = $25\%$

To Find:

The Selling Price (SP) of the music CD.

Solution:

Method 1 (Calculating Discount Amount - Similar to Sonia's approach):

First, calculate the amount of the discount.

Discount Amount = Discount Percentage $\times$ Original Price

The Selling Price is the Original Price minus the Discount Amount.

Method 2 (Calculating Percentage Paid - Similar to Rahul's approach):

If there is a $25\%$ discount, the customer pays $100\% - 25\% = 75\%$ of the Original Price.

Selling Price = Percentage Paid $\times$ Original Price

Both methods show that the Selling Price is $\textsf{₹} 90$.

Answer:

The Selling Price (S.P.) of the music CD is $\textsf{₹} 90$.

We need to compare the sale price of the video game at Store A and Store B.

Given:

Original Price of the video game at both stores = $\textsf{₹} 750$

Sale Price at Store B = $\textsf{₹} 600$

Discount Percentage at Store A = $25\%$ off the original price.

To Find:

At which store the video game is less expensive.

Solution:

The sale price at Store B is directly given as $\textsf{₹} 600$.

Now, let's calculate the sale price at Store A.

The original price at Store A is $\textsf{₹} 750$, and there is a $25\%$ discount.

Discount Amount at Store A = $25\%$ of Original Price

Sale Price at Store A = Original Price - Discount Amount

Alternate Calculation for Sale Price at Store A:

If there is a $25\%$ discount, the customer pays $100\% - 25\% = 75\%$ of the Original Price.

Now, compare the Sale Prices at the two stores:

Sale Price at Store A = $\textsf{₹} 562.50$

Sale Price at Store B = $\textsf{₹} 600.00$

The video game is less expensive at Store A.

Answer:

The game is less expensive at Store A.

The problem states that the sale price of any toy is $66\%$ of its original price.

Given:

Sale Price = $66\%$ of Original Price

Original price of a specific toy = $\textsf{₹} 90$

To Find:

(a) The Sale Price of the toy.

(b) The amount of money saved on the toy.

Solution:

(a) What is the sale price of a toy that originally costs Rs 90?

Sale Price = $66\%$ of Original Price

The sale price of the toy is $\textsf{₹} 59.40$.

(b) How much money would you save on a toy costing Rs 90?

The amount saved is the difference between the Original Price and the Sale Price.

Substitute the values:

Perform the subtraction:

You would save $\textsf{₹} 30.60$ on a toy costing $\textsf{₹} 90$.

Answer:

(a) The sale price of the toy is $\textsf{₹} 59.40$.

(b) You would save $\textsf{₹} 30.60$ on a toy costing $\textsf{₹} 90$.

Let the marked price of the board game be MP.

Given:

Marked Price (MP) = $\textsf{₹} 320$

Discount Percentage = $25\%$

Gift Voucher Value = $\textsf{₹} 50$

To Find:

Answers to parts (a), (b), and (c).

Solution:

a. Do you think both the methods will give the same result? If not, predict which method will be beneficial for her.

No, both methods will likely not give the same result because the order of applying the operations (subtracting the voucher and taking the percentage discount) is different. Applying a percentage discount on a smaller value (after subtracting the voucher) will result in a smaller discount amount compared to applying it on the original larger value.

Let's analyze the steps:

Method 1: $(320 - 50) \times (1 - 0.25) = 270 \times 0.75$

Method 2: $(320 \times (1 - 0.25)) - 50 = (320 \times 0.75) - 50$

Since $270 \times 0.75$ and $(320 \times 0.75) - 50$ involve different base amounts for the percentage calculation, the results will differ.

To predict which is beneficial for her (resulting in a lower final price), consider that Method 2 applies the percentage discount to the full price before subtracting the voucher. This typically results in a larger initial discount amount based on the marked price, making the item cheaper before the voucher is applied. Therefore, Method 2 is likely to be more beneficial for her.

Prediction: Both methods will not give the same result. Method 2 will be more beneficial for Sheela.

b. For each method, calculate the amount Sheela would have to pay. Show your work.

Method 1: Subtract Rs 50 from the price and take 25% off the resulting price.

Step 1: Subtract the voucher amount from the marked price.

Step 2: Take 25% off the resulting price ($\textsf{₹} 270$).

Amount of discount = $25\%$ of $\textsf{₹} 270$

Amount to pay = Price after subtracting voucher - Discount

Alternatively, the price after a 25% discount is 75% of the price:

Method 2: Take 25% off the original price and then subtract Rs 50.

Step 1: Take 25% off the original price ($\textsf{₹} 320$).

Amount of discount = $25\%$ of $\textsf{₹} 320$

Price after discount = Original Price - Discount

Alternatively, the price after a 25% discount is 75% of the original price:

Step 2: Subtract the Rs 50 gift voucher from the price after the discount.

Comparison of results:

Amount to pay (Method 1) = $\textsf{₹} 202.50$

Amount to pay (Method 2) = $\textsf{₹} 190.00$

As predicted in part (a), Method 2 results in a lower final price for Sheela.

c. Which method do you think stores actually use? Why?

Stores typically use Method 2. Here's why:

The percentage discount is a reduction from the marked price of the item. Promotions like "% off" apply directly to the sticker price to determine the initial sale price.

Gift vouchers are a form of payment or a fixed value reduction that is applied after the price of the item has been determined by any applicable discounts. They are treated like cash or store credit.

From the store's perspective, applying the discount to the original price ensures that the percentage discount is calculated on the maximum possible value, potentially making the item more attractive at first glance (a bigger initial discount number) while correctly applying the voucher as a final payment adjustment. Applying the percentage discount after subtracting the voucher (Method 1) would mean the store is giving a discount on a price already reduced by the voucher, effectively diminishing the value of the percentage discount based on the item's marked price. This is not how percentage discounts are advertised or typically applied in retail.

Answer:

a. No, both methods will not give the same result. Method 2 will be more beneficial for Sheela.

b. Amount to pay using Method 1: $\textsf{₹} 202.50$

Amount to pay using Method 2: $\textsf{₹} 190.00$

c. Stores actually use Method 2. They calculate the percentage discount on the original price first and then apply the gift voucher.

We need to calculate the total rent for the first two months at each apartment and compare the costs.

Given:

Neelgiri Apartments:

• Regular monthly rent = $\textsf{₹} 6000$
• Discount for the first two months = $20\%$ off the total rent for these two months.

Savana Apartments:

• Regular monthly rent = $\textsf{₹} 7000$
• Discount for the first month = $50\%$ off the first month's rent.

To Find:

1. Which apartment is cheaper for the first two months?

2. By how much is it cheaper?

Solution:

Cost at Neelgiri Apartments for the first two months:

The rent is $\textsf{₹} 6000$ per month.

Regular total rent for two months = $\textsf{₹} 6000 + \textsf{₹} 6000 = \textsf{₹} 12000$.

There is a $20\%$ discount on this total amount.

Discount Amount = $20\%$ of $\textsf{₹} 12000$

Cost for the first two months (Neelgiri) = Regular total rent - Discount Amount

Alternatively, the cost for the first two months is $100\% - 20\% = 80\%$ of the regular total rent.

Cost at Savana Apartments for the first two months:

The rent is $\textsf{₹} 7000$ per month.

First month's rent has a $50\%$ discount.

Discount on first month = $50\%$ of $\textsf{₹} 7000 = 0.50 \times 7000 = \textsf{₹} 3500$.

Cost for the first month = Regular rent - Discount

Alternatively, the cost for the first month is $100\% - 50\% = 50\%$ of the regular rent.

The second month's rent is the regular price, $\textsf{₹} 7000$.

Total cost for the first two months at Savana Apartments = Cost (Month 1) + Cost (Month 2)

Comparison:

Cost for the first two months at Neelgiri Apartments = $\textsf{₹} 9600$

Cost for the first two months at Savana Apartments = $\textsf{₹} 10500$

Neelgiri Apartments will be cheaper for the first two months.

To find out by how much it is cheaper, calculate the difference:

Difference = Cost (Savana) - Cost (Neelgiri)

Perform the subtraction:

Neelgiri Apartments is cheaper by $\textsf{₹} 900$.

Answer:

Neelgiri Apartments will be cheaper for the first two months by $\textsf{₹} 900$.

Let the original amount be $A$.

Explanation:

Consider an amount $A$.

When the amount is increased by $20\%$, the new amount becomes:

This new amount ($1.20 A$) is now the base for the subsequent $20\%$ decrease.

When this new amount ($1.20 A$) is decreased by $20\%$, the final amount becomes:

Comparing the Final Amount ($0.96 A$) with the Original Amount ($A$ or $1.00 A$), we see that:

This demonstrates that a $20\%$ increase followed by a $20\%$ decrease results in an amount less than the original amount. The reason is that the $20\%$ decrease is applied to the increased amount ($1.20 A$), which is larger than the original amount ($A$). Therefore, the actual amount of the decrease ($0.20 \times 1.20 A = 0.24 A$) is larger than the actual amount of the initial increase ($0.20 \times A = 0.20 A$), leading to a net decrease.

In general, for any non-zero amount, a percentage increase followed by the same percentage decrease will always result in a net decrease. The net change is given by $\left(1 + \frac{x}{100}\right) \left(1 - \frac{x}{100}\right) - 1 = 1 - \left(\frac{x}{100}\right)^2 - 1 = - \left(\frac{x}{100}\right)^2$. Since $\left(\frac{x}{100}\right)^2$ is always positive for $x > 0$, the net change is negative, indicating a decrease.

For a $20\%$ change ($x=20$):

This corresponds to a $4\%$ decrease ($0.04 \times 100\% = 4\%$), resulting in a final amount that is $100\% - 4\% = 96\%$ of the original amount.

Given:

SPF tells you how many minutes you can stay in the sun before receiving one minute of burning UV rays.

SPF X means you receive 1 minute of UV rays for every X minutes in the sun, implying the sunscreen allows $\frac{1}{X}$ of the UV rays.

To Find:

Answers to parts 1, 2(a), 2(b), and 3.

Solution:

1. A sunscreen with SPF 15 allows only $\frac{1}{15}$ of the sun’s UV rays. What per cent of UV rays does the sunscreen abort?

If the sunscreen allows $\frac{1}{15}$ of the UV rays, the fraction of UV rays it blocks (or aborts) is the total amount of rays (represented by 1) minus the amount allowed.

To convert this fraction to a percentage, multiply by $100\%$.

As a decimal percentage:

2. Suppose a sunscreen allows 25% of the sun’s UV rays.

a. What fraction of UV rays does this sunscreen block? Give your answer in lowest terms.

If the sunscreen allows $25\%$ of the UV rays, the percentage it blocks is $100\% - 25\% = 75\%$.

To convert $75\%$ to a fraction in lowest terms:

Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is $25$:

The fraction of UV rays blocked is $\frac{3}{4}$.

b. Use your answer from Part (a) to calculate this sunscreen’s SPF. Explain how you found your answer.

If the sunscreen blocks $\frac{3}{4}$ of the UV rays, the fraction it allows is $1 - \frac{3}{4} = \frac{1}{4}$.

We are given that a sunscreen with SPF X allows $\frac{1}{X}$ of the UV rays.

So, we have the fraction allowed = $\frac{1}{\text{SPF}}$.

In this case, the fraction allowed is $\frac{1}{4}$.

By comparing the denominators, we can see that SPF $= 4$.

Alternatively, from the definition, if it allows $25\%$ of the rays, you can stay in the sun for $X$ minutes to receive $1$ minute of burning rays.

The fraction of rays allowed is $\frac{1}{X}$. We are given that the fraction allowed is $25\% = \frac{25}{100} = \frac{1}{4}$.

This means $X = 4$. So, the SPF is 4.

Explanation: I found the fraction of rays allowed by subtracting the blocked fraction from 1. Knowing that the fraction allowed is equal to $\frac{1}{\text{SPF}}$, I set up the equation $\frac{1}{\text{SPF}} = \frac{1}{4}$ and solved for SPF, which is 4.

3. A label on a sunscreen with SPF 30 claims that the sunscreen blocks about 97% of harmful UV rays. Assuming the SPF factor is accurate, is this claim true? Explain.

According to the definition, a sunscreen with SPF 30 allows $\frac{1}{30}$ of the sun's UV rays.

The fraction of UV rays blocked is $1 - \text{Fraction allowed}$.

Now, convert this fraction to a percentage to compare it with the claim.

Perform the division:

Rounding to the nearest whole number or one decimal place, this is approximately $96.7\%$ or $97\%$.

Comparison:

Calculated percentage blocked $\approx 96.7\%$

Claimed percentage blocked = $97\%$

Yes, the claim is true. The calculated percentage of UV rays blocked by an SPF 30 sunscreen (approximately $96.7\%$) is very close to $97\%$. This difference is likely due to rounding in the stated claim or in the calculation.

Answer:

1. The sunscreen with SPF 15 aborts approximately $93.33\%$ (or $\mathbf{280/3\%}$) of UV rays.

2. a. The fraction of UV rays blocked is $\frac{3}{4}$.

2. b. The sunscreen's SPF is $4$. Explanation: Since it allows $25\% = \frac{1}{4}$ of the UV rays, and SPF X allows $\frac{1}{X}$ of the rays, by comparison, $\frac{1}{\text{SPF}} = \frac{1}{4}$, so SPF = 4.

3. Yes, the claim is true. An SPF 30 sunscreen blocks approximately $96.7\%$ of UV rays, which is about $97\%$.

Let the selling price of the property be SP.

Given:

Commission amount = $\textsf{₹} 50,000$

Commission Percentage = $4\%$ of the Selling Price.

To Find:

The Selling Price (SP) of the property.

Solution:

The commission amount is a percentage of the selling price. We can write this as an equation:

Substitute the given values into the equation:

Convert the percentage to a decimal or fraction:

Simplify the fraction $\frac{4}{100}$ to $\frac{1}{25}$.

To find SP, multiply both sides of the equation by 25:

Perform the multiplication:

Thus, the agent sells the property at a price of $\textsf{₹} 12,50,000$.

Verification:

Calculate $4\%$ of $\textsf{₹} 12,50,000$:

$4\%$ of $\textsf{₹} 1250000 = \frac{4}{100} \times 1250000 = 0.04 \times 1250000 = \textsf{₹} 50000$.

This matches the given commission amount.

Answer:

The agent sells the property at a price of $\textsf{₹} 12,50,000$.

Let the original price of tea per kg be $P_0$ and the reduced price be $P_r$. The expenditure is constant at $\textsf{₹} 45$.

Given:

Expenditure = $\textsf{₹} 45$

Price decrease = $15\%$

Extra quantity bought = $2$ kg

To Find:

Reduced Price ($P_r$) and Original Price ($P_0$).

Solution:

The $15\%$ decrease in price on the original expenditure is the amount saved, which allowed buying the extra $2$ kg.

This $\textsf{₹} 6.75$ is the cost of the extra $2$ kg at the reduced price ($P_r$).

Reduced price per kg ($P_r$) = $\frac{6.75}{2} = \textsf{₹} 3.375$

The reduced price is $85\%$ of the original price ($P_r = 0.85 P_0$).

Original price per kg ($P_0$) = $\frac{3.375}{0.85} = \frac{3375}{850} = \frac{135}{34}$

$P_0 = \textsf{₹} \frac{135}{34}$ per kg.

Answer:

The reduced price of tea was $\textsf{₹} 3.375$ per kg.

The original price of tea was $\textsf{₹} \frac{135}{34}$ per kg.

To complete the table, we need to calculate the final percentage for each subject based on the total marks obtained out of the total maximum marks.

Given:

The table shows the marks obtained in Internal Assessment and Examination for each subject, and the calculated Total marks (Obtained/Maximum).

To Find:

The final percentage for each subject based on the Total marks.

Solution:

The final percentage is calculated as: $\text{Percentage} = \frac{\text{Total Marks Obtained}}{\text{Total Maximum Marks}} \times 100\%$

Let's calculate the percentage for each subject:

1. English Literature: Total = 102/125

2. English Language: Total = 113/125

3. Hindi Literature: Total = 85/100

4. Hindi Language: Total = 84/100

5. Mathematics: Total = 130/150

6. Sanskrit: Total = 99/120

7. Physics: Total = 135/150

8. Chemistry: Total = 123/150

9. Biology: Total = 130/150

10. History and Civics: Total = 87/100

11. Geography: Total = 88.5/100

Completed Table:

Comparative Performance Analysis:

By looking at the Final Percentages, Vidit can compare his performance across subjects:

• He performed best in English Language (90.4%) followed closely by Physics (90%).
• He needs improvement in English Literature (81.6%), as it is his lowest percentage. Chemistry (82%) and Sanskrit (82.5%) are also among his lower scores.

The success rate in percentage is calculated by dividing the number of successful attempts by the total number of attempts and multiplying by 100.

Given:

Number of successful baskets = $32$

Total number of attempts = $35$

To Find:

Sita's success rate in percentage.

Solution:

Success Rate (as a fraction) = $\frac{\text{Number of successful baskets}}{\text{Total number of attempts}}$

To convert this fraction to a percentage, multiply by $100\%$.

Simplify the fraction by dividing both numerator and denominator by 5:

Now, perform the division to get a decimal percentage:

Rounding to two decimal places, the success rate is approximately $91.43\%$.

Answer:

Sita's success rate in percentage is $\frac{640}{7}\%$ (or approximately $91.43\%$).

To compare the amount of homework finished, we need to express both amounts as percentages.

Given:

Percentage of homework finished by Neha = $73\%$

Fraction of homework finished by Minakshi = $\frac{5}{8}$

To Find:

Who finished a greater percentage of homework.

Solution:

Neha finished $73\%$ of her homework. This is already in percentage form.

Minakshi finished $\frac{5}{8}$ of her homework. To convert this fraction to a percentage, we multiply by $100\%$.

Simplify the fraction:

So, Minakshi finished $62.5\%$ of her homework.

Now, we compare the percentages:

Neha's percentage = $73\%$

Minakshi's percentage = $62.5\%$

Neha finished a greater percentage of her homework than Minakshi during school hours.

Answer:

Neha must finish a greater per cent of homework, as she finished $73\%$ while Minakshi finished $62.5\%$.

To find the percentage of plant species found in rain forests, we divide the number of species in rain forests by the total number of identified species in the world and multiply by 100.

Given:

Number of identified plant species in rain forests = $90,000$

Total number of identified plant species in the world = $2,50,000$

To Find:

The percentage of world's identified plant species found in rain forests.

Solution:

Percentage of species in rain forests = $\frac{\text{Number of species in rain forests}}{\text{Total number of species}} \times 100\%$

Simplify the fraction by cancelling out common zeros:

Perform the multiplication:

$36\%$ of the world's identified plant species are found in rain forests.

Answer:

$36\%$ of the world’s identified plant species are found in rain forests.

To find the percentage of the floor covered by the carpet, we need to compare the area of the carpet to the area of the room's floor.

Given:

Room dimensions = $6$m $\times$ $3$m (length $\times$ width)

Area covered by the carpet = $8\text{m}^2$

To Find:

The percentage of the floor covered by the carpet.

Solution:

First, calculate the area of the room's floor.

Area of a rectangle = length $\times$ width

Now, we need to find what percentage the carpet's area ($8\text{m}^2$) is of the floor's area ($18\text{m}^2$).

Percentage covered by carpet = $\frac{\text{Area covered by carpet}}{\text{Area of floor}} \times 100\%$

Simplify the fraction $\frac{8}{18}$ by dividing the numerator and denominator by their greatest common divisor, which is 2.

Perform the division to get a decimal percentage:

As a recurring decimal percentage, this is $44.\overline{4}\%$. As a mixed number percentage, it is $44 \frac{4}{9}\%$. Rounding to two decimal places, it is approximately $44.44\%$.

Answer:

$\frac{400}{9}\%$ (or $44.\overline{4}\%$ or approximately $44.44\%$) of the floor is covered by the carpet.

To find the amount of water in Jyoti's weight, we need to calculate $67\%$ of her total body weight.

Given:

Percentage of body weight that is water = $67\%$

Jyoti's total body weight = $56$ kg

To Find:

The amount of Jyoti's weight that is water (in kg).

Solution:

Amount of water in weight = Percentage of water $\times$ Total Body Weight

Convert the percentage to a decimal or fraction:

Perform the multiplication:

So, $37.52$ kg of Jyoti's weight is water.

Answer:

$37.52$ kg of her weight is water.

To find the amount of pure gold in the ring, we need to calculate $58.3\%$ of the total weight of the ring.

Given:

Percentage of pure gold in 14 carat gold = $58.3\%$

Total weight of the 14 carat gold ring = $7.6$ grams

To Find:

The amount of pure gold in the ring (in grams).

Solution:

Amount of pure gold = Percentage of pure gold $\times$ Total weight of the ring

Convert the percentage to a decimal:

Perform the multiplication:

We can round the answer to a suitable number of decimal places, for example, two decimal places, as the original weights are given with one decimal place.

Answer:

There are approximately $4.43$ grams of pure gold in the ring.

Explanation of the Student's Error:

The proportion used by the student is $\frac{n}{100} = \frac{5}{32}$.

This proportion represents the relationship: $\frac{\text{Part}}{\text{Whole}} = \frac{\text{Percentage}}{100}$.

When finding a percentage of a number, the percentage value should be in the numerator of the percentage fraction ($\frac{5}{100}$ in this case), and the whole amount should be in the denominator of the part-to-whole fraction. The unknown ($n$) represents the part we are trying to find.

The correct proportion to find $5\%$ of $32$ should be:

Let's compare this to the student's proportion $\frac{n}{100} = \frac{5}{32}$.

The student incorrectly placed the whole amount ($32$) in the denominator of the percentage fraction and placed $100$ (which is part of the percentage structure) in the denominator of the part-to-whole fraction.

In simpler terms, the student set up the proportion to find what percentage 5 is of 32, not to find 5% of 32.

If we solve the student's proportion $\frac{n}{100} = \frac{5}{32}$:

This calculates what percentage $5$ is of $32$ (which is $15.625\%$), not $5\%$ of $32$.

To find $5\%$ of $32$ correctly using proportion $\frac{n}{32} = \frac{5}{100}$:

The correct answer for $5\%$ of $32$ is $1.6$.

Answer:

The student set up the proportion incorrectly. The proportion $\frac{n}{100} = \frac{5}{32}$ is used to find what percentage $5$ is of $32$. To find $5\%$ of $32$, the correct proportion should be $\frac{\text{Part (n)}}{\text{Whole (32)}} = \frac{\text{Percentage (5)}}{100}$, which is $\frac{n}{32} = \frac{5}{100}$. The student swapped the positions of $100$ and $32$ in the denominators relative to the standard proportion formula for finding a percentage of a number.

We need to calculate the sales tax rate for each brand and then compare them.

Given:

• Brand X: Cost without tax = $\textsf{₹} 70$, Cost with tax = $\textsf{₹} 75$
• Brand Y: Cost without tax = $\textsf{₹} 62$, Cost with tax = $\textsf{₹} 65$

To Find:

1. The sales tax rate for each brand.

2. Which brand has a greater sales tax rate.

Solution:

The sales tax amount is the difference between the cost with tax and the cost without tax. The sales tax rate is calculated as a percentage of the cost without tax.

Sales Tax Amount = (Cost + Tax) - Cost

Sales Tax Rate $= \frac{\text{Sales Tax Amount}}{\text{Cost (without tax)}} \times 100\%$

For Brand X:

Cost without tax = $\textsf{₹} 70$

Cost with tax = $\textsf{₹} 75$

Sales Tax Amount (X) = $75 - 70 = \textsf{₹} 5$

As a decimal percentage:

Sales Tax Rate (X) $\approx 7.14\%$ (rounded to two decimal places).

For Brand Y:

Cost without tax = $\textsf{₹} 62$

Cost with tax = $\textsf{₹} 65$

Sales Tax Amount (Y) = $65 - 62 = \textsf{₹} 3$

As a decimal percentage:

Sales Tax Rate (Y) $\approx 4.84\%$ (rounded to two decimal places).

Comparison:

Sales Tax Rate (X) $\approx 7.14\%$

Sales Tax Rate (Y) $\approx 4.84\%$

Brand X has a greater sales tax rate.

Answer:

Brand X has a greater sales tax rate.

Sales tax rate of Brand X: $\frac{50}{7}\%$ (or approximately $7.14\%$)

Sales tax rate of Brand Y: $\frac{150}{31}\%$ (or approximately $4.84\%$)