Chapter 8 Comparing Quantities (Additional Questions)

Given:

Quantity 1 = 50 paise

Quantity 2 = $\textsf{₹ }5$

To Find:

The ratio of 50 paise to $\textsf{₹ }5$ in simplest form.

Solution:

To find the ratio between two quantities, they must be in the same unit.

Convert $\textsf{₹ }5$ into paise.

We know that $1$ Rupee $= 100$ paise.

So, $\textsf{₹ }5 = 5 \times 100$ paise.

$\textsf{₹ }5 = 500$ paise.

Now, the ratio of 50 paise to $\textsf{₹ }5$ is the ratio of 50 paise to 500 paise.

Ratio $= \frac{50 \text{ paise}}{500 \text{ paise}}$

To simplify the ratio, divide both the numerator and the denominator by their greatest common divisor (GCD).

The GCD of 50 and 500 is 50.

Ratio $= \frac{\cancel{50}^{1}}{\cancel{500}_{10}}$

Ratio $= \frac{1}{10}$

The ratio can be written as $1:10$.

Comparing the simplified ratio with the given options, we find that option (A) matches our result.

The final answer is (A) 1:10.

Given:

Ratio is $3:4$.

To Find:

Convert the ratio $3:4$ to percentage.

Solution:

A ratio $a:b$ can be written as a fraction $\frac{a}{b}$.

To convert a fraction to a percentage, we multiply the fraction by $100\%$.

Given ratio is $3:4$.

Write the ratio as a fraction:

Fraction $= \frac{3}{4}$

Convert the fraction to a percentage:

Percentage $= \frac{3}{4} \times 100\%$

Now, calculate the value:

Percentage $= 3 \times \frac{\cancel{100}^{25}}{\cancel{4}_{1}}\%$

Percentage $= 3 \times 25\%$

Percentage $= 75\%$

Comparing the result with the given options, we find that option (B) matches our result.

The final answer is (B) $75\%$.

Given:

Total number of students in the class = 50

Number of girls = 30

The rest are boys.

To Find:

The ratio of boys to girls.

Solution:

First, find the number of boys in the class.

Number of boys = Total number of students - Number of girls

Number of boys = $50 - 30$

Number of boys = 20

Now, we need to find the ratio of boys to girls.

Ratio of boys to girls = $\frac{\text{Number of boys}}{\text{Number of girls}}$

Ratio $= \frac{20}{30}$

Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor (GCD).

The GCD of 20 and 30 is 10.

Ratio $= \frac{\cancel{20}^{2}}{\cancel{30}_{3}}$

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

The ratio of boys to girls in the simplest form is $2:3$.

Comparing the result with the given options, we find that option (B) matches our result.

The final answer is (B) 2:3.

Given:

The proportion is $2:5 :: x:15$.

To Find:

The value of $x$ in the given proportion.

Solution:

A proportion states that two ratios are equal.

The given proportion $2:5 :: x:15$ can be written as the equation of two fractions:

$\frac{2}{5} = \frac{x}{15}$

In a proportion, the product of the extremes is equal to the product of the means.

The extremes are the first and last terms (2 and 15).

The means are the middle terms (5 and $x$).

So, we have:

$2 \times 15 = 5 \times x$

$30 = 5x$

To find the value of $x$, divide both sides of the equation by 5:

$\frac{30}{5} = \frac{5x}{5}$

Simplify the fractions:

$\frac{\cancel{30}^{6}}{\cancel{5}_{1}} = \frac{\cancel{5}x}{\cancel{5}}$

$6 = x$

So, the value of $x$ is 6.

Comparing the result with the given options, we find that option (B) matches our result.

The final answer is (B) 6.

Given:

Cost of 5 pens = $\textsf{₹ }60$

To Find:

The cost of 1 pen.

Solution:

We are given the cost of a certain number of pens and asked to find the cost of a single pen. This can be solved using the unitary method.

If the cost of 5 pens is $\textsf{₹ }60$, then the cost of 1 pen is the total cost divided by the number of pens.

Cost of 1 pen $= \frac{\text{Total cost}}{\text{Number of pens}}$

Cost of 1 pen $= \frac{\textsf{₹ }60}{5}$

Calculate the value:

Cost of 1 pen $= \textsf{₹ } \frac{\cancel{60}^{12}}{\cancel{5}_{1}}$

Cost of 1 pen $= \textsf{₹ }12$

Comparing the result with the given options, we find that option (B) matches our result.

The final answer is (B) $\textsf{₹ }12$.

Given:

Cost of 5 pens = $\textsf{₹ }60$ (from Question 5)

Number of pens to find the cost for = 8

To Find:

The cost of 8 pens using the unitary method.

Solution:

The unitary method involves first finding the value of a single unit, and then using that value to find the value of the desired number of units.

From Question 5, we found the cost of 1 pen.

Cost of 1 pen $= \frac{\text{Cost of 5 pens}}{5}$

Cost of 1 pen $= \frac{\textsf{₹ }60}{5}$

Cost of 1 pen $= \textsf{₹ }12$

Now, to find the cost of 8 pens, multiply the cost of 1 pen by the number of pens (8).

Cost of 8 pens $= \text{Cost of 1 pen} \times 8$

Cost of 8 pens $= \textsf{₹ }12 \times 8$

Cost of 8 pens $= \textsf{₹ }96$

Comparing the result with the given options, we find that option (B) matches our result.

The final answer is (B) $\textsf{₹ }96$.

Given:

The decimal is $0.25$.

To Find:

Convert $0.25$ into percentage.

Solution:

To convert a decimal into a percentage, we multiply the decimal by $100\%$.

Percentage $= 0.25 \times 100\%$

When multiplying a decimal by 100, move the decimal point two places to the right.

$0.25 \times 100 = 25$

So, the percentage is $25\%$.

Comparing the result with the given options, we find that option (C) matches our result.

The final answer is (C) $25\%$.

Given:

Percentage = $15\%$

Number = 200

To Find:

Calculate $15\%$ of 200.

Solution:

To find a percentage of a number, we convert the percentage to a fraction or a decimal and then multiply it by the number.

We can convert $15\%$ to a fraction by dividing by 100:

$15\% = \frac{15}{100}$

Now, we need to find $\frac{15}{100}$ of 200. The word "of" in mathematics usually means multiplication.

$15\%$ of $200 = \frac{15}{100} \times 200$

We can simplify the multiplication by cancelling out common factors between the numerator and the denominator.

$15\%$ of $200 = \frac{15}{\cancel{100}_{1}} \times \cancel{200}^{2}$

$15\%$ of $200 = 15 \times 2$

$15\%$ of $200 = 30$

Comparing the result with the given options, we find that option (C) matches our result.

The final answer is (C) 30.

Given:

Cost Price (CP) of the article = $\textsf{₹ }500$

Selling Price (SP) of the article = $\textsf{₹ }550$

To Find:

The profit earned by the shopkeeper.

Solution:

Profit is earned when the Selling Price (SP) of an article is greater than its Cost Price (CP).

In this case, SP = $\textsf{₹ }550$ and CP = $\textsf{₹ }500$. Since $550 > 500$, there is a profit.

The formula for calculating profit is:

Profit = Selling Price (SP) - Cost Price (CP)

Substitute the given values into the formula:

Profit $= \textsf{₹ }550 - \textsf{₹ }500$

Profit $= \textsf{₹ }50$

Comparing the result with the given options, we find that option (A) matches our result.

The final answer is (A) $\textsf{₹ }50$.

Given:

Cost Price (CP) of the article = $\textsf{₹ }500$ (from Question 9)

Selling Price (SP) of the article = $\textsf{₹ }550$ (from Question 9)

Profit = $\textsf{₹ }50$ (from Question 9)

To Find:

The profit percentage.

Solution:

The profit percentage is calculated on the Cost Price (CP).

The formula for Profit Percentage is:

$\text{Profit Percentage} = \left( \frac{\text{Profit}}{\text{CP}} \right) \times 100\%$

Substitute the values of Profit and CP:

$\text{Profit Percentage} = \left( \frac{\textsf{₹ }50}{\textsf{₹ }500} \right) \times 100\%$

$\text{Profit Percentage} = \frac{50}{500} \times 100\%$

Simplify the fraction:

$\text{Profit Percentage} = \frac{\cancel{50}^{1}}{\cancel{500}_{10}} \times 100\%$

$\text{Profit Percentage} = \frac{1}{10} \times 100\%$

$\text{Profit Percentage} = 0.1 \times 100\%$

$\text{Profit Percentage} = 10\%$

Comparing the result with the given options, we find that option (B) matches our result.

The final answer is (B) $10\%$.

Given:

Cost Price (CP) of mangoes = $\textsf{₹ }800$

Selling Price (SP) of mangoes = $\textsf{₹ }700$

To Find:

The loss incurred by the fruit seller.

Solution:

Loss occurs when the Selling Price (SP) of an article is less than its Cost Price (CP).

In this case, SP = $\textsf{₹ }700$ and CP = $\textsf{₹ }800$. Since $700 < 800$, there is a loss.

The formula for calculating loss is:

Loss = Cost Price (CP) - Selling Price (SP)

Substitute the given values into the formula:

Loss $= \textsf{₹ }800 - \textsf{₹ }700$

Loss $= \textsf{₹ }100$

Comparing the result with the given options, we find that option (A) matches our result.

The final answer is (A) $\textsf{₹ }100$.

Given:

Cost Price (CP) of mangoes = $\textsf{₹ }800$ (from Question 11)

Selling Price (SP) of mangoes = $\textsf{₹ }700$ (from Question 11)

Loss = $\textsf{₹ }100$ (from Question 11)

To Find:

The loss percentage.

Solution:

The loss percentage is calculated on the Cost Price (CP).

The formula for Loss Percentage is:

$\text{Loss Percentage} = \left( \frac{\text{Loss}}{\text{CP}} \right) \times 100\%$

Substitute the values of Loss and CP:

$\text{Loss Percentage} = \left( \frac{\textsf{₹ }100}{\textsf{₹ }800} \right) \times 100\%$

$\text{Loss Percentage} = \frac{100}{800} \times 100\%$

Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor (GCD), which is 100.

$\text{Loss Percentage} = \frac{\cancel{100}^{1}}{\cancel{800}_{8}} \times 100\%$

$\text{Loss Percentage} = \frac{1}{8} \times 100\%$

Now, calculate the value:

$\text{Loss Percentage} = \frac{100}{8}\%$

Divide 100 by 8:

$\text{Loss Percentage} = 12.5\%$

Comparing the result with the given options, we find that option (B) matches our result.

The final answer is (B) $12.5\%$.

Given:

Principal ($P$) = $\textsf{₹ }1000$

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

Time ($T$) = 2 years

To Find:

The Simple Interest (SI).

Solution:

The formula for calculating Simple Interest is:

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

Substitute the given values into the formula:

$SI = \frac{1000 \times 5 \times 2}{100}$

Calculate the product in the numerator:

$1000 \times 5 \times 2 = 1000 \times 10 = 10000$

So, the formula becomes:

$SI = \frac{10000}{100}$

Perform the division:

$SI = \frac{\cancel{10000}^{100}}{\cancel{100}_{1}}$

$SI = 100$

The Simple Interest is $\textsf{₹ }100$.

Comparing the result with the given options, we find that option (B) matches our result.

The final answer is (B) $\textsf{₹ }100$.

Given:

Principal ($P$) = $\textsf{₹ }1000$ (from Question 13)

Simple Interest (SI) = $\textsf{₹ }100$ (calculated in Question 13)

To Find:

The total amount to be repaid after 2 years.

Solution:

The total amount to be repaid is the sum of the principal and the simple interest.

Amount = Principal + Simple Interest

Substitute the given values:

Amount $= \textsf{₹ }1000 + \textsf{₹ }100$

Amount $= \textsf{₹ }1100$

So, the total amount to be repaid after 2 years is $\textsf{₹ }1100$.

Comparing the result with the given options, we find that option (B) matches our result.

The final answer is (B) $\textsf{₹ }1100$.

Given:

A list of numbers or ratios in different forms.

To Match:

Match the equivalent values from the two columns.

Solution:

We need to convert the items in the left column to match one of the forms in the right column.

Let's convert each item in the left column:

(i) $\frac{1}{2}$

Convert to percentage: $\frac{1}{2} \times 100\% = 50\%$. This matches option (c).

Convert to decimal: $\frac{1}{2} = 0.5$. This does not match any decimal in the right column.

So, (i) matches (c).

(ii) $0.75$

Convert to fraction: $0.75 = \frac{75}{100} = \frac{\cancel{75}^{3}}{\cancel{100}_{4}} = \frac{3}{4}$. This matches option (d).

Convert to percentage: $0.75 \times 100\% = 75\%$. This does not match any percentage in the right column.

So, (ii) matches (d).

(iii) $20\%$

Convert to decimal: $20\% = \frac{20}{100} = 0.2$. This matches option (a).

Convert to fraction: $20\% = \frac{20}{100} = \frac{\cancel{20}^{1}}{\cancel{100}_{5}} = \frac{1}{5}$. This does not match any fraction in the right column.

So, (iii) matches (a).

(iv) $1:4$

Convert to fraction: $1:4 = \frac{1}{4}$. This does not match any fraction in the right column.

Convert to percentage: $\frac{1}{4} \times 100\% = 25\%$. This matches option (b).

Convert to decimal: $\frac{1}{4} = 0.25$. This does not match any decimal in the right column.

So, (iv) matches (b).

The matches are:

(i) - (c)

(ii) - (d)

(iii) - (a)

(iv) - (b)

Let's check the given options to find the one that matches our results.

Option (A) is (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b).

This matches our calculated pairings.

The final answer is (A) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b).

Given:

Assertion (A): If two quantities are in proportion, their ratio must be equal.

Reason (R): Proportion is an equality of two ratios.

To Determine:

Whether Assertion (A) and Reason (R) are true, and if Reason (R) is the correct explanation for Assertion (A).

Solution:

Let's analyze the truthfulness of each statement.

Reason (R): "Proportion is an equality of two ratios."

By definition, a proportion is a statement that two ratios are equal. For example, if four quantities $a, b, c, d$ are in proportion, it is written as $a:b :: c:d$, which means $\frac{a}{b} = \frac{c}{d}$. This statement is the standard definition of proportion.

Therefore, Reason (R) is true.

Assertion (A): "If two quantities are in proportion, their ratio must be equal."

This statement refers to the ratios involved in a proportion. A proportion typically involves four quantities forming two ratios. For example, if quantities $a, b, c, d$ are in proportion, it means the ratio $a:b$ is equal to the ratio $c:d$. The assertion states that the ratios that form the proportion are equal, which is consistent with the definition of proportion.

Therefore, Assertion (A) is true, assuming it refers to the ratios that constitute the proportion.

Now, let's consider if Reason (R) is the correct explanation for Assertion (A).

Assertion (A) states a property of quantities in proportion: that their ratios are equal. Reason (R) provides the definition of proportion: it *is* the equality of two ratios. The definition (R) directly explains *why* the ratios are equal when quantities are in proportion (A). Assertion (A) is essentially a direct consequence or restatement of the definition given in Reason (R).

Thus, Reason (R) is the correct explanation for Assertion (A).

Based on the analysis:

Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).

Comparing the conclusion with the given options, we find that option (A) matches our result.

The final answer is (A) Both A and R are true, and R is the correct explanation of A.

Given:

Assertion (A): If the Cost Price (CP) of an article is $\textsf{₹ }200$ and Selling Price (SP) is $\textsf{₹ }250$, there is a profit of $\textsf{₹ }50$.

Reason (R): Profit = CP - SP.

To Determine:

Whether Assertion (A) and Reason (R) are true, and if Reason (R) is the correct explanation for Assertion (A).

Solution:

Let's analyze the truthfulness of each statement.

Reason (R): "Profit = CP - SP."

The formula for Profit is when the Selling Price (SP) is greater than the Cost Price (CP). In that case, Profit = SP - CP. The formula given in Reason (R), CP - SP, is the formula for Loss when CP is greater than SP.

Therefore, Reason (R) is false.

Assertion (A): "If the Cost Price (CP) of an article is $\textsf{₹ }200$ and Selling Price (SP) is $\textsf{₹ }250$, there is a profit of $\textsf{₹ }50$."

Here, CP = $\textsf{₹ }200$ and SP = $\textsf{₹ }250$.

Since SP ($250$) is greater than CP ($200$), there is a profit.

Profit = SP - CP

Profit $= \textsf{₹ }250 - \textsf{₹ }200$

Profit $= \textsf{₹ }50$

The statement in Assertion (A) is consistent with the calculation.

Therefore, Assertion (A) is true.

Since Reason (R) is false, it cannot be the correct explanation for Assertion (A).

Based on the analysis:

Assertion (A) is true, but Reason (R) is false.

Comparing the conclusion with the given options, we find that option (C) matches our result.

The final answer is (C) A is true, but R is false.

Given:

Cost Price (CP) of the dress = $\textsf{₹ }1200$

Loss Percentage = $20\%$

To Find:

The loss amount.

Solution:

The loss amount is calculated as a percentage of the Cost Price (CP).

Formula for Loss Amount = $\frac{\text{Loss Percentage}}{100} \times \text{CP}$

Substitute the given values into the formula:

Loss Amount $= \frac{20}{100} \times \textsf{₹ }1200$

Simplify the fraction $\frac{20}{100}$:

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

So, the calculation becomes:

Loss Amount $= \frac{1}{5} \times \textsf{₹ }1200$

Calculate the value:

Loss Amount $= \textsf{₹ } \frac{\cancel{1200}^{240}}{\cancel{5}_{1}}$

Loss Amount $= \textsf{₹ }240$

So, the loss amount is $\textsf{₹ }240$.

Comparing the result with the given options, we find that option (C) matches our result.

The final answer is (C) $\textsf{₹ }240$.

Given:

Cost Price (CP) of the dress = $\textsf{₹ }1200$ (from Question 18 case study)

Loss Percentage = $20\%$ (from Question 18 case study)

Loss Amount = $\textsf{₹ }240$ (calculated in the solution to Question 18)

To Find:

The Selling Price (SP) of the dress.

Solution:

When there is a loss, the Selling Price (SP) is calculated by subtracting the Loss Amount from the Cost Price (CP).

The formula is:

Selling Price (SP) = Cost Price (CP) - Loss Amount

Substitute the given values:

SP $= \textsf{₹ }1200 - \textsf{₹ }240$

Perform the subtraction:

SP $= \textsf{₹ }960$

So, the selling price of the dress was $\textsf{₹ }960$.

Comparing the result with the given options, we find that option (C) matches our result.

The final answer is (C) $\textsf{₹ }960$.

Given:

Formula fragment for Simple Interest: Principal $\times$ Rate $\times$ Time / ______.

To Find:

The number that completes the formula for Simple Interest.

Solution:

The standard formula for calculating Simple Interest (SI) is:

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

where:

$P$ is the Principal amount.

$R$ is the Rate of Interest per annum (usually expressed as a percentage, which is why we divide by 100).

$T$ is the Time period in years.

Comparing the given formula fragment "Principal $\times$ Rate $\times$ Time / ______" with the standard formula $\frac{P \times R \times T}{100}$, we can see that the blank should be filled with the number 100.

So, the complete formula is: Simple Interest = Principal $\times$ Rate $\times$ Time / 100.

Comparing the required number with the given options, we find that option (B) matches our result.

The final answer is (B) 100.

Given:

Two conditions relating Cost Price (CP) and Selling Price (SP).

To Find:

Fill in the blanks with the appropriate terms (Profit or Loss).

Solution:

Let's analyze the conditions:

Condition 1: If CP < SP

This means the Selling Price is greater than the Cost Price. When an article is sold for more than it was bought, the difference is called a **Profit**.

Profit = SP - CP (when SP > CP)

So, the first blank should be Profit.

Condition 2: If CP > SP

This means the Cost Price is greater than the Selling Price. When an article is sold for less than it was bought, the difference is called a **Loss**.

Loss = CP - SP (when CP > SP)

So, the second blank should be Loss.

Putting the terms into the blanks, the completed statement is: If CP < SP, there is a Profit. If CP > SP, there is a Loss.

Comparing this with the given options, we find that option (B) matches our conclusion.

The final answer is (B) Profit, Loss.

Given:

The fraction is $\frac{4}{5}$.

To Find:

Convert the fraction $\frac{4}{5}$ into percentage.

Solution:

To convert a fraction into a percentage, we multiply the fraction by $100\%$.

Percentage $= \frac{4}{5} \times 100\%$

Now, we calculate the value:

Percentage $= 4 \times \frac{\cancel{100}^{20}}{\cancel{5}_{1}}\%$

Percentage $= 4 \times 20\%$

Percentage $= 80\%$

So, the fraction $\frac{4}{5}$ is equivalent to $80\%$.

Comparing the result with the given options, we find that option (C) matches our result.

The final answer is (C) $80\%$.

Given:

Scored marks = 36

Total marks = 50

To Find:

The percentage score.

Solution:

To find the percentage score, we divide the marks obtained by the total marks and multiply by $100\%$.

Percentage Score $= \left( \frac{\text{Scored Marks}}{\text{Total Marks}} \right) \times 100\%$

Substitute the given values:

Percentage Score $= \left( \frac{36}{50} \right) \times 100\%$

Calculate the value:

Percentage Score $= \frac{36}{\cancel{50}_{1}} \times \cancel{100}^{2}\%$

Percentage Score $= 36 \times 2\%$

Percentage Score $= 72\%$

So, your percentage score is $72\%$.

Comparing the result with the given options, we find that option (C) matches our result.

The final answer is (C) $72\%$.

Given:

The target ratio is $1:2$.

The options are (A) $2:4$, (B) $10:20$, and (C) $5:10$.

To Find:

Which of the given ratios are equivalent to $1:2$.

Solution:

Two ratios are equivalent if their simplified forms are the same.

The target ratio is $1:2$, which can be written as the fraction $\frac{1}{2}$. This is already in its simplest form.

Now, let's check each option:

(A) $2:4$

Write as a fraction: $\frac{2}{4}$.

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2.

$\frac{\cancel{2}^{1}}{\cancel{4}_{2}} = \frac{1}{2}$

The simplified ratio is $1:2$. This is equivalent to the target ratio.

(B) $10:20$

Write as a fraction: $\frac{10}{20}$.

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 10.

$\frac{\cancel{10}^{1}}{\cancel{20}_{2}} = \frac{1}{2}$

The simplified ratio is $1:2$. This is equivalent to the target ratio.

(C) $5:10$

Write as a fraction: $\frac{5}{10}$.

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 5.

$\frac{\cancel{5}^{1}}{\cancel{10}_{2}} = \frac{1}{2}$

The simplified ratio is $1:2$. This is equivalent to the target ratio.

Since ratios (A), (B), and (C) are all equivalent to $1:2$, the correct option is (D) All of the above.

The final answer is (D) All of the above.

Given:

Cost of 12 books = $\textsf{₹ }300$

To Find:

The cost of 7 books.

Solution:

We can use the unitary method to solve this problem. First, find the cost of 1 book.

Cost of 1 book $= \frac{\text{Cost of 12 books}}{\text{Number of books}}$

Cost of 1 book $= \frac{\textsf{₹ }300}{12}$

Divide 300 by 12:

Cost of 1 book $= \textsf{₹ } \frac{\cancel{300}^{25}}{\cancel{12}_{1}}$

Cost of 1 book $= \textsf{₹ }25$

Now that we know the cost of 1 book, we can find the cost of 7 books by multiplying the cost of 1 book by 7.

Cost of 7 books $= \text{Cost of 1 book} \times 7$

Cost of 7 books $= \textsf{₹ }25 \times 7$

Cost of 7 books $= \textsf{₹ }175$

Comparing the result with the given options, we find that option (B) matches our result.

The final answer is (B) $\textsf{₹ }175$.

Given:

Original Price of petrol = $\textsf{₹ }90$ per litre

Percentage Increase in price = $10\%$

To Find:

The new price of petrol per litre.

Solution:

First, calculate the amount of increase in the price of petrol.

The increase amount is $10\%$ of the original price.

Increase Amount $= 10\%$ of $\textsf{₹ }90$

To calculate a percentage of a number, we convert the percentage to a fraction (by dividing by 100) and multiply it by the number.

Increase Amount $= \frac{10}{100} \times \textsf{₹ }90$

Simplify the fraction $\frac{10}{100}$:

$\frac{10}{100} = \frac{\cancel{10}^{1}}{\cancel{100}_{10}} = \frac{1}{10}$

So, the Increase Amount calculation is:

Increase Amount $= \frac{1}{10} \times \textsf{₹ }90$

Increase Amount $= \textsf{₹ } \frac{90}{10}$

Increase Amount $= \textsf{₹ }9$

Now, to find the new price, add the increase amount to the original price.

New Price = Original Price + Increase Amount

New Price $= \textsf{₹ }90 + \textsf{₹ }9$

New Price $= \textsf{₹ }99$

So, the new price of petrol is $\textsf{₹ }99$ per litre.

Comparing the result with the given options, we find that option (A) matches our result.

The final answer is (A) $\textsf{₹ }99$.

Given:

Principal ($P$) = $\textsf{₹ }5000$

Rate of Interest ($R$) = $6\%$ per annum

Time ($T$) = 3 years

To Find:

The Simple Interest (SI) earned.

Solution:

The formula for calculating Simple Interest is:

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

Substitute the given values into the formula:

$SI = \frac{5000 \times 6 \times 3}{100}$

Calculate the product in the numerator:

$5000 \times 6 = 30000$

$30000 \times 3 = 90000$

So, the formula becomes:

$SI = \frac{90000}{100}$

Perform the division:

$SI = \frac{\cancel{90000}^{900}}{\cancel{100}_{1}}$

$SI = 900$

The total simple interest earned is $\textsf{₹ }900$.

Comparing the result with the given options, we find that option (B) matches our result.

The final answer is (B) $\textsf{₹ }900$.

Given:

Four options, each representing a potential proportion.

To Find:

Identify which of the given options represents a proportion.

Solution:

A proportion is an equality between two ratios. To check if a given statement represents a proportion, we need to verify if the two ratios are equal.

Let's check each option:

(A) $2:3 = 4:5$

This represents the equality of the ratios $\frac{2}{3}$ and $\frac{4}{5}$.

To check if $\frac{2}{3} = \frac{4}{5}$, we can cross-multiply or find a common denominator.

Using cross-multiplication: $2 \times 5 = 10$ and $3 \times 4 = 12$.

Since $10 \neq 12$, the ratios are not equal.

So, $2:3 = 4:5$ is not a proportion.

(B) $1:5 = 3:15$

This represents the equality of the ratios $\frac{1}{5}$ and $\frac{3}{15}$.

To check if $\frac{1}{5} = \frac{3}{15}$, we can simplify the second ratio or cross-multiply.

Simplifying the second ratio: $\frac{3}{15} = \frac{\cancel{3}^{1}}{\cancel{15}_{5}} = \frac{1}{5}$.

Since $\frac{1}{5} = \frac{1}{5}$, the ratios are equal.

So, $1:5 = 3:15$ is a proportion.

(C) $6:8 = 3:5$

This represents the equality of the ratios $\frac{6}{8}$ and $\frac{3}{5}$.

Simplify the first ratio: $\frac{6}{8} = \frac{\cancel{6}^{3}}{\cancel{8}_{4}} = \frac{3}{4}$.

Now we check if $\frac{3}{4} = \frac{3}{5}$.

Using cross-multiplication: $3 \times 5 = 15$ and $4 \times 3 = 12$.

Since $15 \neq 12$, the ratios are not equal.

So, $6:8 = 3:5$ is not a proportion.

(D) $10:1 = 5:0.5$

This represents the equality of the ratios $\frac{10}{1}$ and $\frac{5}{0.5}$.

The first ratio is $\frac{10}{1} = 10$.

The second ratio is $\frac{5}{0.5}$. To remove the decimal, multiply the numerator and denominator by 10: $\frac{5 \times 10}{0.5 \times 10} = \frac{50}{5} = 10$.

Since $10 = 10$, the ratios are equal.

So, $10:1 = 5:0.5$ is a proportion.

Both options (B) and (D) represent proportions. However, in a multiple-choice question with a single correct answer, option (B) uses whole numbers in both ratios and is a more common example of a simple proportion. Option (D) involves a decimal, which is also valid but might be considered less fundamental as an example of a proportion unless the question specifically deals with decimals. Assuming a single correct answer from the provided options, option (B) is the most likely intended answer.

Comparing the results with the given options, option (B) is a proportion.

The final answer is (B) $1:5 = 3:15$.

Given:

The fraction is $\frac{1}{8}$.

To Find:

Express the fraction $\frac{1}{8}$ as a percentage.

Solution:

To convert a fraction into a percentage, we multiply the fraction by $100\%$.

Percentage $= \frac{1}{8} \times 100\%$

Now, we calculate the value:

Percentage $= \frac{100}{8}\%$

We can simplify the fraction or perform the division.

$\frac{100}{8} = \frac{\cancel{100}^{25}}{\cancel{8}_{2}} = \frac{25}{2}$

Percentage $= \frac{25}{2}\%$

Converting the fraction to a decimal:

$\frac{25}{2} = 12.5$

So, the percentage is $12.5\%$.

Comparing the result with the given options, we find that option (C) matches our result.

The final answer is (C) $12.5\%$.

Given:

Cost of a dozen bananas = $\textsf{₹ }48$.

We know that one dozen is equal to 12 items.

So, the cost of 12 bananas = $\textsf{₹ }48$.

To Find:

The cost of 5 bananas.

Solution:

We can use the unitary method to solve this problem. First, we find the cost of a single banana.

Cost of 1 banana $= \frac{\text{Cost of 12 bananas}}{12}$

Cost of 1 banana $= \frac{\textsf{₹ }48}{12}$

Divide 48 by 12:

Cost of 1 banana $= \textsf{₹ }4$

Now that we know the cost of 1 banana, we can find the cost of 5 bananas by multiplying the cost of 1 banana by the desired number of bananas (5).

Cost of 5 bananas $= \text{Cost of 1 banana} \times 5$

Cost of 5 bananas $= \textsf{₹ }4 \times 5$

Cost of 5 bananas $= \textsf{₹ }20$

So, the cost of 5 bananas is $\textsf{₹ }20$.

Comparing the result with the given options, we find that option (B) matches our result.

The final answer is (B) $\textsf{₹ }20$.

Given:

Ratio of red to blue marbles = $4:5$.

Number of red marbles = 20.

To Find:

The number of blue marbles.

Solution:

The ratio of red to blue marbles is given as $4:5$. This means that the number of red marbles is proportional to 4 parts, and the number of blue marbles is proportional to 5 parts.

We are given that the number of red marbles is 20.

So, we can equate the number of red marbles to the corresponding parts in the ratio:

4 parts = 20 marbles

To find the value of one part, divide the number of red marbles by the number of parts for red marbles:

Value of 1 part $= \frac{20}{4}$ marbles

Value of 1 part $= 5$ marbles.

Now, we need to find the number of blue marbles. The ratio shows that the number of blue marbles corresponds to 5 parts.

Number of blue marbles $= \text{Number of blue parts in the ratio} \times \text{Value of 1 part}$

Number of blue marbles $= 5 \times 5$ marbles

Number of blue marbles $= 25$ marbles.

Thus, there are 25 blue marbles in the bag.

Comparing the result with the given options, we find that option (C) matches our result.

The final answer is (C) 25.

Given:

Original Price $= \textsf{₹ }200$

New Price $= \textsf{₹ }240$

To Find:

Percentage Increase

Solution:

First, we find the amount of increase in the price.

Increase in Price $= \text{New Price} - \text{Original Price}$

Increase in Price $= \textsf{₹ }240 - \textsf{₹ }200$

Increase in Price $= \textsf{₹ }40$

Now, we calculate the percentage increase using the formula:

Percentage Increase $= \frac{\text{Increase}}{\text{Original Price}} \times 100\%$

Substitute the values:

Percentage Increase $= \frac{\textsf{₹ }40}{\textsf{₹ }200} \times 100\%$

Percentage Increase $= \frac{40}{200} \times 100\%$

We can simplify the fraction:

Percentage Increase $= \frac{\cancel{40}^{1}}{\cancel{200}_{5}} \times 100\%$

Percentage Increase $= \frac{1}{5} \times 100\%$

Percentage Increase $= \frac{100}{5} \%$

Percentage Increase $= 20\%$

Thus, the percentage increase in price is $\textbf{20\%}$.

Comparing this result with the given options, we find that option (A) is correct.

The correct option is (A) $20\%$.

Given:

Principal Amount $(P) = \textsf{₹ }8000$

Time Period $(T) = 3 \text{ years}$

Simple Interest $(SI) = \textsf{₹ }1200$

To Find:

Rate of Interest $(R)$ per annum.

Solution:

The formula for Simple Interest is:

We need to find the Rate of Interest $(R)$. We can rearrange the formula to solve for $R$:

Now, substitute the given values into this formula:

$R = \frac{\textsf{₹ }1200 \times 100}{\textsf{₹ }8000 \times 3 \text{ years}}$

$R = \frac{1200 \times 100}{8000 \times 3}$

$R = \frac{120000}{24000}$

To simplify, we can cancel out the zeros:

$R = \frac{120 \cancel{00}}{24 \cancel{00}}$

$R = \frac{120}{24}$

Divide 120 by 24:

$R = 5$

The rate of interest is $5\%$ per annum.

Comparing this result with the given options, we find that option (B) is correct.

The correct option is (B) $5\%$.

Let's analyze each statement about ratios:

(A) The two quantities in a ratio must be of the same unit. This statement is TRUE. Ratios compare quantities of the same kind. If the units are different (e.g., comparing length in meters to length in centimeters), they must be converted to the same unit before forming the ratio.

(B) The order of terms in a ratio is important. This statement is TRUE. The ratio $a:b$ is generally not the same as the ratio $b:a$. For example, the ratio of 2 apples to 3 oranges ($2:3$) is different from the ratio of 3 oranges to 2 apples ($3:2$).

(C) A ratio has no unit. This statement is TRUE. Since the quantities being compared are of the same unit, the units cancel out when the ratio is formed. For example, the ratio of 10 cm to 5 cm is $\frac{10 \text{ cm}}{5 \text{ cm}} = \frac{10}{5} = 2$, which is a unitless number.

(D) A ratio can be negative. This statement is FALSE. Ratios typically involve the comparison of the magnitudes of quantities, which are non-negative. While individual components might represent negative values in some contexts (like coordinates), the concept of a ratio in the fundamental sense (comparing amounts, lengths, weights, etc.) uses non-negative values. A ratio like $a:b$ where $a$ and $b$ are positive represents a positive comparison.

Therefore, the statement that is FALSE is that a ratio can be negative.

The correct option is (D) A ratio can be negative.

Given:

Cost Price (CP) $= \textsf{₹ }400$

Profit $= \textsf{₹ }80$

To Find:

Selling Price (SP)

Solution:

The Selling Price (SP) is the sum of the Cost Price (CP) and the Profit.

The formula is:

$SP = CP + \text{Profit}$

Substitute the given values into the formula:

$SP = \textsf{₹ }400 + \textsf{₹ }80$

$SP = \textsf{₹ }480$

Therefore, the Selling Price of the article is $\textsf{₹ }480$.

Comparing this result with the given options, we find that option (B) is correct.

The correct option is (B) ₹ 480.

Given:

Fraction $= \frac{3}{8}$

To Find:

Percentage equivalent of the given fraction.

Solution:

To convert a fraction into a percentage, we multiply the fraction by $100\%$.

Percentage $= \text{Fraction} \times 100\%$

Substitute the given fraction:

Percentage $= \frac{3}{8} \times 100\%$

Now, calculate the value:

Percentage $= \frac{3 \times 100}{8} \%$

Percentage $= \frac{300}{8} \%$

Perform the division:

$\frac{300}{8} = \frac{\cancel{300}^{75}}{\cancel{8}_{2}} = \frac{75}{2} = 37.5$

So, Percentage $= 37.5\%$

Thus, the fraction $\frac{3}{8}$ converted into percentage is $\textbf{37.5\%}$.

Comparing this result with the given options, we find that option (B) is correct.

The correct option is (B) $37.5\%$.

Given:

Principal $(P) = \textsf{₹ }10000$

Rate of Interest $(R) = 10\%$ per annum

Time Period $(T) = 1 \text{ year}$

To Find:

Simple Interest $(SI)$

Solution:

The formula for calculating Simple Interest is:

Substitute the given values into the formula (i):

$SI = \frac{10000 \times 10 \times 1}{100}$

$SI = \frac{100000}{100}$

$SI = 1000$

The Simple Interest is $\textsf{₹ }1000$.

Comparing this result with the given options, we find that option (B) is correct.

The correct option is (B) ₹ 1000.

We need to find the ratio of the number of days in a week to the number of months in a year.

Number of days in a week $= 7$.

Number of months in a year $= 12$.

The ratio is given by:

Ratio $= (\text{Number of days in a week}) : (\text{Number of months in a year})$

Ratio $= 7 : 12$

The ratio of the number of days in a week to the number of months in a year is $\textbf{7:12}$.

Comparing this result with the given options, we find that option (A) is correct.

The correct option is (A) 7:12.

Given:

Distance covered in 3 hours $= 150\text{ km}$

Time taken $= 3\text{ hours}$

New Time $= 5\text{ hours}$

Speed is constant.

To Find:

Distance covered in 5 hours.

Solution:

First, calculate the speed of the car using the given distance and time.

The formula for speed is:

$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$

Substitute the initial values:

$\text{Speed} = \frac{150\text{ km}}{3\text{ hours}}$

$\text{Speed} = 50\text{ km/hour}$

The car travels at a speed of $50\text{ km/hour}$.

Now, use this speed to find the distance covered in 5 hours.

The formula for distance is:

$\text{Distance} = \text{Speed} \times \text{Time}$

Substitute the calculated speed and the new time:

$\text{Distance} = 50\text{ km/hour} \times 5\text{ hours}$

$\text{Distance} = 250\text{ km}$

Thus, the car will cover a distance of $\textbf{250 km}$ in 5 hours at the same speed.

Comparing this result with the given options, we find that option (B) is correct.

The correct option is (B) $250\text{ km}$.

Given:

Percentage $= 60\%$

To Find:

Percentage as a fraction in simplest form, from the given options.

Solution:

To convert a percentage to a fraction, we divide the percentage value by 100.

$60\% = \frac{60}{100}$

Now, we need to simplify this fraction to its simplest form. We can divide both the numerator and the denominator by their greatest common divisor (GCD).

The greatest common divisor (GCD) of 60 and 100 is 20.

Divide the numerator and denominator by 20:

$\frac{60}{100} = \frac{60 \div 20}{100 \div 20} = \frac{3}{5}$

The fraction $\frac{3}{5}$ is in simplest form because the only common factor of 3 and 5 is 1.

Let's examine the given options:

(A) $\frac{60}{100}$: This is a valid fraction for $60\%$, but it is not in simplest form.

(B) $\frac{6}{10}$: This is also a valid fraction for $60\%$. We can simplify it as $\frac{6}{10} = \frac{6 \div 2}{10 \div 2} = \frac{3}{5}$. It is simpler than $\frac{60}{100}$, but not the simplest form listed.

(C) $\frac{3}{5}$: This fraction is in simplest form, as 3 and 5 have no common factors other than 1. It represents $60\%$.

(D) All are equivalent, but (C) is simplest form: This statement correctly describes the relationship between options (A), (B), and (C). All three fractions $\frac{60}{100}$, $\frac{6}{10}$, and $\frac{3}{5}$ are equivalent and represent $60\%$. Among these, $\frac{3}{5}$ (option C) is indeed the simplest form.

Considering the options, option (D) provides the most complete and accurate information that answers the question regarding the simplest form among the given choices and their equivalence.

The correct option is (D) All are equivalent, but (C) is simplest form.

Given:

First quantity = $2$ meters

Second quantity = $40$ cm

To Find:

The ratio of $2$ meters to $40$ cm in its simplest form.

Solution:

To find the ratio of two quantities, their units must be the same.

We will convert meters to centimeters.

We know that $1$ meter = $100$ cm.

So, $2$ meters = $2 \times 100$ cm = $200$ cm.

Now, the ratio of $2$ meters to $40$ cm is the ratio of $200$ cm to $40$ cm.

Ratio $= \frac{\text{First quantity}}{\text{Second quantity}}$

Ratio $= \frac{200 \text{ cm}}{40 \text{ cm}}$

Ratio $= \frac{200}{40}$

To write the ratio in its simplest form, we divide the numerator and the denominator by their greatest common divisor (GCD).

We can simplify the fraction $\frac{200}{40}$ by cancelling common factors:

$\frac{200}{40} = \frac{\cancel{200}^{20}}{\cancel{40}_{4}}$ (Dividing numerator and denominator by $10$)

$= \frac{20}{4}$

Now, divide by $4$:

$\frac{20}{4} = \frac{\cancel{20}^{5}}{\cancel{4}_{1}}$

$= \frac{5}{1}$

The ratio $\frac{5}{1}$ can be written as $5:1$.

Thus, the ratio of $2$ meters to $40$ cm is $5:1$.

The ratio of $2$ meters to $40$ cm in its simplest form is $5:1$.

Given:

Number of boys in the class = $25$

Number of girls in the class = $20$

To Find:

The ratio of the number of boys to the total number of students in the class.

Solution:

First, we need to find the total number of students in the class.

Total number of students = Number of boys + Number of girls

Total number of students = $25 + 20$

Total number of students = $45$

Now, we need to find the ratio of the number of boys to the total number of students.

Ratio $= \frac{\text{Number of boys}}{\text{Total number of students}}$

Ratio $= \frac{25}{45}$

To express the ratio in its simplest form, we find the greatest common divisor (GCD) of the numerator ($25$) and the denominator ($45$).

The factors of $25$ are $1, 5, 25$.

The factors of $45$ are $1, 3, 5, 9, 15, 45$.

The GCD of $25$ and $45$ is $5$.

Divide both the numerator and the denominator by their GCD, which is $5$:

Ratio $= \frac{25 \div 5}{45 \div 5}$

Ratio $= \frac{5}{9}$

Alternatively, we can simplify the fraction by cancelling common factors:

$\frac{25}{45} = \frac{\cancel{25}^{5}}{\cancel{45}_{9}}$

$= \frac{5}{9}$

The ratio $\frac{5}{9}$ can be written in the form $a:b$ as $5:9$.

Thus, the ratio of the number of boys to the total number of students is $5:9$.

The ratio of the number of boys to the total number of students in the class is $5:9$.

Given:

Ratio 1: $15:20$

Ratio 2: $18:24$

To Determine:

Whether the ratios $15:20$ and $18:24$ are equivalent.

Solution:

Two ratios are equivalent if their simplest forms are the same.

First, let's find the simplest form of the ratio $15:20$.

Ratio $15:20$ can be written as the fraction $\frac{15}{20}$.

To simplify the fraction, we find the greatest common divisor (GCD) of $15$ and $20$.

Factors of $15$: $1, 3, 5, 15$

Factors of $20$: $1, 2, 4, 5, 10, 20$

The GCD of $15$ and $20$ is $5$.

Divide the numerator and the denominator by $5$:

$\frac{15}{20} = \frac{15 \div 5}{20 \div 5} = \frac{3}{4}$

So, the simplest form of the ratio $15:20$ is $3:4$.

Next, let's find the simplest form of the ratio $18:24$.

Ratio $18:24$ can be written as the fraction $\frac{18}{24}$.

To simplify the fraction, we find the greatest common divisor (GCD) of $18$ and $24$.

Factors of $18$: $1, 2, 3, 6, 9, 18$

Factors of $24$: $1, 2, 3, 4, 6, 8, 12, 24$

The GCD of $18$ and $24$ is $6$.

Divide the numerator and the denominator by $6$:

$\frac{18}{24} = \frac{18 \div 6}{24 \div 6} = \frac{3}{4}$

So, the simplest form of the ratio $18:24$ is $3:4$.

Comparing the simplest forms of both ratios:

Simplest form of $15:20$ is $3:4$.

Simplest form of $18:24$ is $3:4$.

Since the simplest forms of both ratios are the same ($3:4$), the ratios $15:20$ and $18:24$ are equivalent.

The ratios $15:20$ and $18:24$ are equivalent.

Given:

The four numbers are $4, 6, 8, 12$.

To Determine:

Whether the numbers $4, 6, 8, 12$ are in proportion using the product of extremes and means.

Solution:

Four numbers $a, b, c, d$ are in proportion if $a:b :: c:d$. This relationship can be expressed as $\frac{a}{b} = \frac{c}{d}$.

In a proportion $a:b :: c:d$, the first and fourth terms ($a$ and $d$) are called the extremes, and the second and third terms ($b$ and $c$) are called the means.

A fundamental property of proportion states that the product of the extremes is equal to the product of the means. That is, if $a, b, c, d$ are in proportion, then $a \times d = b \times c$.

Given the numbers $4, 6, 8, 12$, we assume they are in proportion in this order:

$4:6 :: 8:12$

Here, the extreme terms are $4$ and $12$.

The mean terms are $6$ and $8$.

Calculate the product of the extremes:

Product of extremes $= 4 \times 12 = 48$

Calculate the product of the means:

Product of means $= 6 \times 8 = 48$

Now, we compare the product of the extremes and the product of the means:

Product of extremes $= 48$

Product of means $= 48$

Since the product of the extremes is equal to the product of the means ($48 = 48$), the numbers $4, 6, 8, 12$ are in proportion.

Reasoning: The numbers $4, 6, 8, 12$ are in proportion because when arranged as $4:6 :: 8:12$, the product of the extreme terms ($4 \times 12 = 48$) is equal to the product of the mean terms ($6 \times 8 = 48$). This equality confirms that the ratios $\frac{4}{6}$ and $\frac{8}{12}$ are equivalent, and therefore the numbers are in proportion.

The numbers $4, 6, 8, 12$ are in proportion.

Given:

The four numbers $6, x, 15, 25$ are in proportion.

To Find:

The value of $x$.

Solution:

If four numbers $a, b, c, d$ are in proportion, then $a:b :: c:d$. This means the ratio of the first two numbers is equal to the ratio of the last two numbers: $\frac{a}{b} = \frac{c}{d}$.

In a proportion, the product of the extremes (the first and fourth terms) is equal to the product of the means (the second and third terms).

Given numbers in proportion are $6, x, 15, 25$.

So, we can write the proportion as:

$6 : x :: 15 : 25$

Applying the property that the product of extremes equals the product of means:

(First term) $\times$ (Fourth term) = (Second term) $\times$ (Third term)

Product of extremes = $6 \times 25$

Product of means = $x \times 15$

Setting them equal:

Now, we solve for $x$ from equation (i):

$6 \times 25 = 150$

So, the equation becomes:

$150 = 15x$

To find $x$, we divide both sides by $15$:

$x = \frac{150}{15}$

$x = 10$

Thus, the value of $x$ that makes the numbers $6, x, 15, 25$ in proportion is $10$.

The value of $x$ is $10$.

Given:

Cost of $5$ kg of sugar = $\textsf{₹}200$

To Find:

The cost of $1$ kg of sugar using the unitary method.

Solution:

The unitary method involves finding the value of a single unit first.

We are given the cost of $5$ kg of sugar is $\textsf{₹}200$.

To find the cost of $1$ kg of sugar, we need to divide the total cost by the quantity of sugar.

Cost of $1$ kg of sugar $= \frac{\text{Total cost}}{\text{Quantity of sugar}}$

Cost of $1$ kg of sugar $= \frac{\textsf{₹}200}{5 \text{ kg}}$

Now, we calculate the division:

Cost of $1$ kg of sugar $= \textsf{₹}\frac{200}{5}$

$= \textsf{₹}40$

So, the cost of $1$ kg of sugar is $\textsf{₹}40$.

Using the unitary method, the cost of $1$ kg of sugar is $\textsf{₹}40$.

Solution:

To convert a fraction to a percentage, we multiply the fraction by $100\%$.

(a) Convert $\frac{3}{4}$ to a percentage:

Percentage $= \frac{3}{4} \times 100\%$

We can simplify this calculation:

Percentage $= \frac{3}{\cancel{4}_{1}} \times \cancel{100}^{25}\%$

Percentage $= 3 \times 25\%$

Percentage $= 75\%$

The fraction $\frac{3}{4}$ is equal to $75\%$.

(b) Convert $\frac{7}{10}$ to a percentage:

Percentage $= \frac{7}{10} \times 100\%$

We can simplify this calculation:

Percentage $= \frac{7}{\cancel{10}_{1}} \times \cancel{100}^{10}\%$

Percentage $= 7 \times 10\%$

Percentage $= 70\%$

The fraction $\frac{7}{10}$ is equal to $70\%$.

Solution:

To convert a decimal to a percentage, we multiply the decimal by $100\%$.

(a) Convert $0.25$ to a percentage:

Percentage $= 0.25 \times 100\%$

$0.25 \times 100 = 25$

Percentage $= 25\%$

The decimal $0.25$ is equal to $25\%$.

(b) Convert $1.5$ to a percentage:

Percentage $= 1.5 \times 100\%$

$1.5 \times 100 = 150$

Percentage $= 150\%$

The decimal $1.5$ is equal to $150\%$.

Solution:

To convert a percentage to a fraction, we write the percentage value as the numerator and $100$ as the denominator, and then simplify the fraction to its simplest form.

Percentage $P\%$ can be written as the fraction $\frac{P}{100}$.

(a) Convert $40\%$ to a fraction:

$40\% = \frac{40}{100}$

Now, we simplify the fraction $\frac{40}{100}$ by dividing the numerator and the denominator by their greatest common divisor (GCD).

The GCD of $40$ and $100$ is $20$.

$\frac{40}{100} = \frac{40 \div 20}{100 \div 20} = \frac{2}{5}$

The percentage $40\%$ is equal to the fraction $\frac{2}{5}$ in its simplest form.

(b) Convert $75\%$ to a fraction:

$75\% = \frac{75}{100}$

Now, we simplify the fraction $\frac{75}{100}$ by dividing the numerator and the denominator by their greatest common divisor (GCD).

The GCD of $75$ and $100$ is $25$.

$\frac{75}{100} = \frac{75 \div 25}{100 \div 25} = \frac{3}{4}$

The percentage $75\%$ is equal to the fraction $\frac{3}{4}$ in its simplest form.

Solution:

To convert a percentage to a decimal, we divide the percentage value by $100$.

Percentage $P\%$ can be written as the decimal $\frac{P}{100}$.

(a) Convert $18\%$ to a decimal:

Decimal $= \frac{18}{100}$

To divide by $100$, we move the decimal point two places to the left.

$18$ has a decimal point after the $8$ (i.e., $18.0$).

Moving the decimal point two places left gives $0.18$.

Decimal $= 0.18$

The percentage $18\%$ is equal to the decimal $0.18$.

(b) Convert $120\%$ to a decimal:

Decimal $= \frac{120}{100}$

To divide by $100$, we move the decimal point two places to the left.

$120$ has a decimal point after the $0$ (i.e., $120.0$).

Moving the decimal point two places left gives $1.20$ or $1.2$.

Decimal $= 1.2$

The percentage $120\%$ is equal to the decimal $1.2$.

Given:

Percentage = $20\%$

Quantity = $\textsf{₹}800$

To Find:

$20\%$ of $\textsf{₹}800$.

Solution:

To find a percentage of a quantity, we can convert the percentage to a fraction or a decimal and then multiply it by the quantity.

Method 1: Converting percentage to fraction

$20\% = \frac{20}{100}$

$20\%$ of $\textsf{₹}800 = \frac{20}{100} \times \textsf{₹}800$

$= \frac{20}{\cancel{100}_{1}} \times \cancel{800}^{8}\textsf{₹}$

$= 20 \times 8\textsf{₹}$

$= 160\textsf{₹}$

Method 2: Converting percentage to decimal

$20\% = \frac{20}{100} = 0.20$ or $0.2$

$20\%$ of $\textsf{₹}800 = 0.2 \times \textsf{₹}800$

$0.2 \times 800 = 160$

$= 160\textsf{₹}$

Both methods give the same result.

$20\%$ of $\textsf{₹}800$ is $\textsf{₹}160$.

$20\%$ of $\textsf{₹}800$ is $\textsf{₹}160$.

Given:

Percentage = $15\%$

Quantity = $250$ kg

To Find:

$15\%$ of $250$ kg.

Solution:

To find a percentage of a quantity, we can convert the percentage to a fraction or a decimal and then multiply it by the quantity.

Method 1: Converting percentage to fraction

$15\% = \frac{15}{100}$

Now, calculate $15\%$ of $250$ kg:

$15\%$ of $250$ kg $= \frac{15}{100} \times 250$ kg

$= \frac{15 \times 250}{100}$ kg

We can simplify the fraction by dividing the numerator and denominator by common factors. Let's divide by $50$:

$= \frac{15 \times \cancel{250}^{5}}{\cancel{100}_{2}}$ kg

$= \frac{15 \times 5}{2}$ kg

$= \frac{75}{2}$ kg

$= 37.5$ kg

Method 2: Converting percentage to decimal

$15\% = \frac{15}{100} = 0.15$

Now, calculate $15\%$ of $250$ kg:

$15\%$ of $250$ kg $= 0.15 \times 250$ kg

Performing the multiplication:

$0.15 \times 250 = 37.5$

$= 37.5$ kg

Both methods yield the same result.

$15\%$ of $250$ kg is $37.5$ kg.

$15\%$ of $250$ kg is $37.5$ kg.

Given:

Cost Price (CP) of the bicycle = $\textsf{₹}3000$

Selling Price (SP) of the bicycle = $\textsf{₹}3500$

To Find:

The profit made by the shopkeeper.

Solution:

Profit occurs when the Selling Price (SP) is greater than the Cost Price (CP).

In this case, $\textsf{₹}3500 > \textsf{₹}3000$, so there is a profit.

The formula for calculating profit is:

Profit = Selling Price (SP) - Cost Price (CP)

Substitute the given values into the formula:

Profit $= \textsf{₹}3500 - \textsf{₹}3000$

Profit $= \textsf{₹}500$

The profit made by the shopkeeper is $\textsf{₹}500$.

The profit made by the shopkeeper is $\textsf{₹}500$.

Given:

Cost Price (CP) of the article = $\textsf{₹}500$

Selling Price (SP) of the article = $\textsf{₹}450$

To Find:

The loss incurred.

Solution:

Loss occurs when the Selling Price (SP) is less than the Cost Price (CP).

In this case, $\textsf{₹}450 < \textsf{₹}500$, so there is a loss.

The formula for calculating loss is:

Loss = Cost Price (CP) - Selling Price (SP)

Substitute the given values into the formula:

Loss $= \textsf{₹}500 - \textsf{₹}450$

Loss $= \textsf{₹}50$

The loss incurred is $\textsf{₹}50$.

The loss incurred is $\textsf{₹}50$.

Given:

Cost Price (CP) of the article = $\textsf{₹}400$

Selling Price (SP) of the article = $\textsf{₹}500$

To Find:

The profit percentage.

Solution:

First, we need to find the profit made.

Profit = Selling Price (SP) - Cost Price (CP)

Profit $= \textsf{₹}500 - \textsf{₹}400$

Profit $= \textsf{₹}100$

Now, we calculate the profit percentage. The profit percentage is calculated on the Cost Price.

Profit Percentage $= \frac{\text{Profit}}{\text{Cost Price (CP)}} \times 100\%$

Substitute the values of Profit and CP:

Profit Percentage $= \frac{\textsf{₹}100}{\textsf{₹}400} \times 100\%$

Profit Percentage $= \frac{100}{400} \times 100\%$

Simplify the expression:

Profit Percentage $= \frac{\cancel{100}^{1}}{\cancel{400}_{4}} \times 100\%$

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

Profit Percentage $= \frac{100}{4}\%$

Profit Percentage $= 25\%$

The profit percentage is $25\%$.

The profit percentage is $25\%$.

Given:

Cost Price (CP) of apples per kg = $\textsf{₹}60$

Selling Price (SP) of apples per kg = $\textsf{₹}54$

To Find:

The loss percentage.

Solution:

First, we need to find the loss incurred per kg.

Loss occurs when the Selling Price (SP) is less than the Cost Price (CP).

In this case, $\textsf{₹}54 < \textsf{₹}60$, so there is a loss.

Loss = Cost Price (CP) - Selling Price (SP)

Loss $= \textsf{₹}60 - \textsf{₹}54$

Loss $= \textsf{₹}6$

Now, we calculate the loss percentage. The loss percentage is calculated on the Cost Price.

Loss Percentage $= \frac{\text{Loss}}{\text{Cost Price (CP)}} \times 100\%$

Substitute the values of Loss and CP:

Loss Percentage $= \frac{\textsf{₹}6}{\textsf{₹}60} \times 100\%$

Loss Percentage $= \frac{6}{60} \times 100\%$

Simplify the expression:

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

Loss Percentage $= \frac{1}{10} \times 100\%$

Loss Percentage $= \frac{100}{10}\%$

Loss Percentage $= 10\%$

The loss percentage is $10\%$.

The loss percentage is $10\%$.

Given:

Principal (P) = $\textsf{₹}10,000$

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

Time (T) = $2$ years

To Find:

The Simple Interest (SI).

Solution:

The formula for calculating Simple Interest is:

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

Substitute the given values into the formula:

$SI = \frac{10000 \times 5 \times 2}{100}$

Calculate the numerator and simplify:

$SI = \frac{10000 \times 10}{100}$

$SI = \frac{100000}{100}$

Divide the numerator by the denominator:

$SI = 1000$

The Simple Interest is $\textsf{₹}1000$.

The Simple Interest on $\textsf{₹}10,000$ at $5\%$ per annum for $2$ years is $\textsf{₹}1000$.

Given:

Principal (P) = $\textsf{₹}5,000$

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

Time (T) = $1$ year

To Find:

The Simple Interest (SI).

Solution:

The formula for calculating Simple Interest is:

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

Substitute the given values into the formula:

$SI = \frac{5000 \times 8 \times 1}{100}$

Calculate the numerator:

$5000 \times 8 \times 1 = 5000 \times 8 = 40000$

So, the equation becomes:

$SI = \frac{40000}{100}$

Divide the numerator by the denominator:

$SI = 400$

The Simple Interest is $\textsf{₹}400$.

The simple interest is $\textsf{₹}400$.

Given:

Marks scored by the student = $450$

Maximum marks in the exam = $500$

To Find:

The percentage of marks obtained by the student.

Solution:

To find the percentage of marks obtained, we use the formula:

Percentage of marks $= \frac{\text{Marks Scored}}{\text{Maximum Marks}} \times 100\%$

Substitute the given values into the formula:

Percentage of marks $= \frac{450}{500} \times 100\%$

Simplify the expression:

Percentage of marks $= \frac{450}{\cancel{500}_{5}} \times \cancel{100}^{1}\%$

Percentage of marks $= \frac{450}{5}\%$

Percentage of marks $= 90\%$

Alternatively, simplify the fraction first:

$\frac{450}{500} = \frac{45}{50} = \frac{9}{10}$

Percentage of marks $= \frac{9}{10} \times 100\%$

Percentage of marks $= \frac{9}{\cancel{10}_{1}} \times \cancel{100}^{10}\%$

Percentage of marks $= 9 \times 10\% = 90\%$

The percentage of marks obtained by the student is $90\%$.

The percentage of marks obtained is $90\%$.

Given:

Original amount = $\textsf{₹}2500$

Percentage increase = $10\%$

To Find:

The new amount after the increase.

Solution:

To find the new amount after an increase, we first calculate the amount of the increase and then add it to the original amount.

Calculate the amount of the increase:

Amount of increase $= 10\%$ of $\textsf{₹}2500$

Convert the percentage to a fraction or decimal:

$10\% = \frac{10}{100}$

Amount of increase $= \frac{10}{100} \times \textsf{₹}2500$

Amount of increase $= \frac{1}{\cancel{10}_{1}} \times \frac{\cancel{2500}^{250}}{\cancel{100}_{10}}\textsf{₹}$

Amount of increase $= \frac{1}{\cancel{10}_{1}} \times \cancel{250}^{25}\textsf{₹}$

Amount of increase $= 1 \times 25\textsf{₹}$

Amount of increase $= \textsf{₹}250$

Now, add the amount of increase to the original amount to find the new amount:

New amount = Original amount + Amount of increase

New amount $= \textsf{₹}2500 + \textsf{₹}250$

New amount $= \textsf{₹}2750$

The new amount after increasing $\textsf{₹}2500$ by $10\%$ is $\textsf{₹}2750$.

The new amount is $\textsf{₹}2750$.

Given:

Original weight = $150$ kg

Percentage decrease = $20\%$

To Find:

The new weight after the decrease.

Solution:

To find the new weight after a decrease, we first calculate the amount of the decrease and then subtract it from the original weight.

Calculate the amount of the decrease:

Amount of decrease $= 20\%$ of $150$ kg

Convert the percentage to a fraction:

$20\% = \frac{20}{100}$

Amount of decrease $= \frac{20}{100} \times 150$ kg

Amount of decrease $= \frac{\cancel{20}^{1}}{\cancel{100}_{5}} \times 150$ kg

Amount of decrease $= \frac{1}{5} \times 150$ kg

Amount of decrease $= \frac{150}{5}$ kg

Amount of decrease $= 30$ kg

Now, subtract the amount of decrease from the original weight to find the new weight:

New weight = Original weight - Amount of decrease

New weight $= 150$ kg $- 30$ kg

New weight $= 120$ kg

The new weight after decreasing $150$ kg by $20\%$ is $120$ kg.

The new weight is $120$ kg.

Given:

Selling Price (SP) of the shirt = $\textsf{₹}600$

Profit made = $\textsf{₹}120$

To Find:

The Cost Price (CP) of the shirt.

Solution:

We know the relationship between Cost Price, Selling Price, and Profit:

Profit = Selling Price (SP) - Cost Price (CP)

To find the Cost Price (CP), we can rearrange the formula:

Cost Price (CP) = Selling Price (SP) - Profit

Substitute the given values into the formula:

CP $= \textsf{₹}600 - \textsf{₹}120$

Perform the subtraction:

CP $= \textsf{₹}480$

The cost price of the shirt is $\textsf{₹}480$.

The cost price of the shirt is $\textsf{₹}480$.

Given:

Principal (P) = $\textsf{₹}12,000$

Rate of Interest (R) = $6\%$ per annum

Time (T) = $3$ years

To Find:

The total Amount (A) to be paid at the end of $3$ years.

Solution:

First, we need to calculate the Simple Interest (SI) accrued over the given period.

The formula for Simple Interest is:

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

Substitute the given values into the formula:

$SI = \frac{12000 \times 6 \times 3}{100}$

Calculate the numerator and simplify:

$SI = \frac{12000 \times 18}{100}$

$SI = \frac{216000}{100}$

$SI = 2160$

The Simple Interest is $\textsf{₹}2160$.

Now, we need to find the total Amount (A) to be paid. The amount is the sum of the Principal and the Simple Interest.

Amount (A) = Principal (P) + Simple Interest (SI)

Substitute the values of Principal and Simple Interest:

A $= \textsf{₹}12000 + \textsf{₹}2160$

Perform the addition:

A $= \textsf{₹}14160$

The total amount to be paid at the end of $3$ years is $\textsf{₹}14160$.

The amount to be paid at the end of $3$ years is $\textsf{₹}14,160$.

Given:

Ratio of Income to Expenditure $= 7:5$

Income = $\textsf{₹}14,000$

To Find:

The Expenditure of the family.

Solution:

Let the income of the family be $7x$ and the expenditure be $5x$, where $x$ is a common multiplier.

We are given that the income is $\textsf{₹}14,000$.

So, we can set up the equation:

$7x = \textsf{₹}14,000$

To find the value of $x$, divide both sides of the equation by $7$:

$x = \frac{\textsf{₹}14,000}{7}$

$x = \textsf{₹}2000$

Now that we have the value of $x$, we can find the expenditure, which is represented by $5x$.

Expenditure $= 5x$

Expenditure $= 5 \times \textsf{₹}2000$

Expenditure $= \textsf{₹}10,000$

Thus, if the income is $\textsf{₹}14,000$, the expenditure is $\textsf{₹}10,000$.

The expenditure is $\textsf{₹}10,000$.

Given:

Total number of students in the school = $800$

Percentage of girls = $40\%$

To Find:

The number of boys in the school.

Solution:

Method 1: Find the number of girls and subtract from the total.

First, calculate the number of girls in the school.

Number of girls $= 40\%$ of $800$

Number of girls $= \frac{40}{100} \times 800$

Number of girls $= \frac{40}{\cancel{100}_{1}} \times \cancel{800}^{8}$

Number of girls $= 40 \times 8$

Number of girls $= 320$

Now, subtract the number of girls from the total number of students to find the number of boys.

Number of boys = Total students - Number of girls

Number of boys $= 800 - 320$

Number of boys $= 480$

Method 2: Find the percentage of boys first.

If $40\%$ of the students are girls, the remaining percentage must be boys.

Percentage of boys $= 100\% - \text{Percentage of girls}$

Percentage of boys $= 100\% - 40\%$

Percentage of boys $= 60\%$

Now, calculate the number of boys, which is $60\%$ of the total students.

Number of boys $= 60\%$ of $800$

Number of boys $= \frac{60}{100} \times 800$

Number of boys $= \frac{60}{\cancel{100}_{1}} \times \cancel{800}^{8}$

Number of boys $= 60 \times 8$

Number of boys $= 480$

Both methods give the same result.

The number of boys in the school is $480$.

The number of boys in the school is $480$.

Given:

Total amount to be divided = $\textsf{₹}1500$

Ratio among A, B, and C = $3:5:7$

To Find:

The share of each person (A, B, and C).

Solution:

The given ratio is $3:5:7$. This means that for every $3$ parts A gets, B gets $5$ parts, and C gets $7$ parts of the total amount.

First, find the sum of the ratio parts:

Sum of ratio parts $= 3 + 5 + 7$

Sum of ratio parts $= 15$

The total amount $\textsf{₹}1500$ is divided into $15$ equal parts.

To find the value of one ratio part, divide the total amount by the sum of the ratio parts:

Value of one ratio part $= \frac{\text{Total amount}}{\text{Sum of ratio parts}}$

Value of one ratio part $= \frac{\textsf{₹}1500}{15}$

Value of one ratio part $= \textsf{₹}100$

Now, calculate the share of each person by multiplying their respective ratio part by the value of one ratio part:

Share of A = Ratio part of A $\times$ Value of one ratio part

Share of A $= 3 \times \textsf{₹}100$

Share of A $= \textsf{₹}300$

Share of B = Ratio part of B $\times$ Value of one ratio part

Share of B $= 5 \times \textsf{₹}100$

Share of B $= \textsf{₹}500$

Share of C = Ratio part of C $\times$ Value of one ratio part

Share of C $= 7 \times \textsf{₹}100$

Share of C $= \textsf{₹}700$

To verify the answer, check if the sum of the shares is equal to the total amount:

Sum of shares $= \textsf{₹}300 + \textsf{₹}500 + \textsf{₹}700$

Sum of shares $= \textsf{₹}1500$

This matches the total amount given.

The share of A is $\textsf{₹}300$, the share of B is $\textsf{₹}500$, and the share of C is $\textsf{₹}700$.

Given:

Cost of $7$ meters of cloth = $\textsf{₹}1050$

To Find:

1. The cost of $12$ meters of cloth.

2. The number of meters of cloth that can be purchased for $\textsf{₹}2100$.

Solution:

We will use the unitary method to solve both parts of the problem.

Part 1: Find the cost of 12 meters of cloth.

Using the unitary method, we first find the cost of $1$ meter of cloth.

Cost of $7$ meters of cloth $= \textsf{₹}1050$

Cost of $1$ meter of cloth $= \frac{\text{Cost of 7 meters}}{\text{Quantity of cloth}}$

Cost of $1$ meter of cloth $= \frac{\textsf{₹}1050}{7}$

Cost of $1$ meter of cloth $= \textsf{₹}150$

Now that we know the cost of $1$ meter, we can find the cost of $12$ meters by multiplying the cost of $1$ meter by $12$.

Cost of $12$ meters of cloth $= \text{Cost of 1 meter} \times 12$

Cost of $12$ meters of cloth $= \textsf{₹}150 \times 12$

Cost of $12$ meters of cloth $= \textsf{₹}1800$

Part 2: Find the number of meters of cloth for ₹2100.

Using the unitary method, we first find the quantity of cloth that can be purchased for $\textsf{₹}1$.

Quantity of cloth for $\textsf{₹}1050 = 7$ meters

Quantity of cloth for $\textsf{₹}1 = \frac{\text{Quantity of cloth for ₹1050}}{\text{Amount of money}}$

Quantity of cloth for $\textsf{₹}1 = \frac{7 \text{ meters}}{1050}$

Quantity of cloth for $\textsf{₹}1 = \frac{\cancel{7}^{1}}{\cancel{1050}_{150}}$ meters

Quantity of cloth for $\textsf{₹}1 = \frac{1}{150}$ meters

Now that we know the quantity of cloth for $\textsf{₹}1$, we can find the quantity for $\textsf{₹}2100$ by multiplying the quantity for $\textsf{₹}1$ by $2100$.

Quantity of cloth for $\textsf{₹}2100 = \text{Quantity of cloth for ₹1} \times 2100$

Quantity of cloth for $\textsf{₹}2100 = \frac{1}{150} \times 2100$ meters

Quantity of cloth for $\textsf{₹}2100 = \frac{2100}{150}$ meters

Quantity of cloth for $\textsf{₹}2100 = \frac{210}{15}$ meters

Quantity of cloth for $\textsf{₹}2100 = \frac{\cancel{210}^{42}}{\cancel{15}_{3}}$ meters

Quantity of cloth for $\textsf{₹}2100 = \frac{42}{3}$ meters

Quantity of cloth for $\textsf{₹}2100 = 14$ meters

The cost of $12$ meters of cloth is $\textsf{₹}1800$.

$14$ meters of cloth can be purchased for $\textsf{₹}2100$.

Given:

New Salary = $\textsf{₹}23,000$

Percentage Increase = $15\%$

To Find:

The Original Salary.

Solution:

Let the original salary be denoted by $S$.

The increase in salary is $15\%$ of the original salary.

Increase amount $= 15\%$ of $S$

Increase amount $= \frac{15}{100} \times S$

Increase amount $= 0.15S$

The new salary is the sum of the original salary and the increase amount.

New Salary = Original Salary + Increase amount

$\textsf{₹}23,000 = S + 0.15S$

$\textsf{₹}23,000 = (1 + 0.15)S$

$\textsf{₹}23,000 = 1.15S$

To find the original salary $S$, we divide the new salary by $1.15$.

$S = \frac{\textsf{₹}23,000}{1.15}$

Alternatively, if the original salary is $100\%$, after a $15\%$ increase, the new salary is $100\% + 15\% = 115\%$ of the original salary.

So, $115\%$ of Original Salary = $\textsf{₹}23,000$

$\frac{115}{100} \times \text{Original Salary} = \textsf{₹}23,000$

$1.15 \times \text{Original Salary} = \textsf{₹}23,000$

Original Salary $= \frac{\textsf{₹}23,000}{1.15}$

Now we calculate the value:

Original Salary $= \frac{23000}{1.15} = \frac{23000}{\frac{115}{100}} = \frac{23000 \times 100}{115}$

Original Salary $= \frac{2300000}{115}$

Since $23000 = 200 \times 115$, the calculation is:

$\frac{23000}{115} = 200$

So, $\frac{23000 \times 100}{115} = 200 \times 100 = 20000$

Original Salary $= \textsf{₹}20,000$

The original salary was $\textsf{₹}20,000$.

The original salary is $\textsf{₹}20,000$.

Given:

Purchase price of the washing machine = $\textsf{₹}12,000$

Expenses on transportation = $\textsf{₹}500$

Expenses on packaging = $\textsf{₹}300$

Selling Price (SP) of the washing machine = $\textsf{₹}14,720$

To Find:

The profit or loss percentage.

Solution:

First, we need to calculate the total Cost Price (CP) of the washing machine. The total CP includes the purchase price and any additional expenses incurred before selling.

Total Cost Price (CP) = Purchase price + Transportation expenses + Packaging expenses

Total CP $= \textsf{₹}12,000 + \textsf{₹}500 + \textsf{₹}300$

Total CP $= \textsf{₹}12,800$

Now, compare the Total Cost Price (CP) and the Selling Price (SP).

Total CP $= \textsf{₹}12,800$

SP $= \textsf{₹}14,720$

Since SP ($ \textsf{₹}14,720$) is greater than CP ($ \textsf{₹}12,800$), there is a profit.

Calculate the amount of profit:

Profit = Selling Price (SP) - Total Cost Price (CP)

Profit $= \textsf{₹}14,720 - \textsf{₹}12,800$

Profit $= \textsf{₹}1,920$

Now, calculate the profit percentage. The profit percentage is always calculated on the Cost Price.

Profit Percentage $= \frac{\text{Profit}}{\text{Total Cost Price (CP)}} \times 100\%$

Substitute the values of Profit and Total CP:

Profit Percentage $= \frac{\textsf{₹}1,920}{\textsf{₹}12,800} \times 100\%$

Profit Percentage $= \frac{1920}{12800} \times 100\%$

Simplify the fraction $\frac{1920}{12800}$:

Divide numerator and denominator by $10$:

$\frac{192}{1280}$

Divide numerator and denominator by their GCD, which is $64$:

$\frac{192 \div 64}{1280 \div 64} = \frac{3}{20}$

Alternatively, using cancellation:

Profit Percentage $= \frac{\cancel{1920}^{192}}{\cancel{12800}_{1280}} \times 100\%$

Profit Percentage $= \frac{\cancel{192}^{3}}{\cancel{1280}_{20}} \times 100\%$

Profit Percentage $= \frac{3}{20} \times 100\%$

Calculate the final percentage:

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

Profit Percentage $= 3 \times 5\%$

Profit Percentage $= 15\%$

The shopkeeper made a profit of $15\%$.

The profit percentage is $15\%$.

Given:

Principal (P) = $\textsf{₹}25,000$

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

Time (T) = $5$ years

To Find:

1. The Simple Interest (SI).

2. The total Amount (A) to be paid at the end of $5$ years.

Solution:

First, we will calculate the Simple Interest (SI) using the formula:

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

Substitute the given values into the formula:

$SI = \frac{25000 \times 12 \times 5}{100}$

Calculate the product in the numerator:

$12 \times 5 = 60$

$SI = \frac{25000 \times 60}{100}$

$SI = \frac{1500000}{100}$

Divide the numerator by the denominator:

$SI = 15000$

The Simple Interest is $\textsf{₹}15,000$.

Next, we will calculate the total Amount (A) to be paid at the end of $5$ years. The amount is the sum of the Principal and the Simple Interest.

Amount (A) = Principal (P) + Simple Interest (SI)

Substitute the values of Principal and Simple Interest:

A $= \textsf{₹}25,000 + \textsf{₹}15,000$

Perform the addition:

A $= \textsf{₹}40,000$

The Simple Interest is $\textsf{₹}15,000$.

The total amount to be paid is $\textsf{₹}40,000$.

Given:

Original Population = $15,000$

New Population = $16,500$

To Find:

The percentage increase in the population.

Solution:

First, calculate the increase in population.

Increase in population = New Population - Original Population

Increase in population $= 16,500 - 15,000$

Increase in population $= 1,500$

Now, calculate the percentage increase. The percentage increase is calculated with respect to the original population.

Percentage Increase $= \frac{\text{Increase in population}}{\text{Original Population}} \times 100\%$

Substitute the values into the formula:

Percentage Increase $= \frac{1500}{15000} \times 100\%$

Simplify the fraction:

Percentage Increase $= \frac{\cancel{1500}^{1}}{\cancel{15000}_{10}} \times 100\%$

Percentage Increase $= \frac{1}{10} \times 100\%$

Percentage Increase $= \frac{100}{10}\%$

Percentage Increase $= 10\%$

The percentage increase in the population is $10\%$.

The percentage increase in the population is $10\%$.

Given:

Selling Price (SP) with loss = $\textsf{₹}19,200$

Loss Percentage = $4\%$

Desired Profit Percentage = $8\%$

To Find:

The selling price required to gain $8\%$.

Solution:

First, we need to find the Cost Price (CP) of the sofa set. We are given the selling price and the loss percentage when sold at that price.

When there is a loss, the Selling Price is a percentage of the Cost Price given by:

$SP = CP \times (1 - \frac{\text{Loss}\%}{100})$

Substitute the given values:

$\textsf{₹}19,200 = CP \times (1 - \frac{4}{100})$

$\textsf{₹}19,200 = CP \times (1 - 0.04)$

$\textsf{₹}19,200 = CP \times 0.96$

Now, solve for CP:

$CP = \frac{\textsf{₹}19,200}{0.96}$

$CP = \frac{19200}{\frac{96}{100}} = \frac{19200 \times 100}{96}$

$CP = \frac{1920000}{96}$

$CP = \textsf{₹}20,000$

So, the Cost Price of the sofa set is $\textsf{₹}20,000$.

Next, we need to find the Selling Price ($SP_{new}$) that would result in an $8\%$ profit on this Cost Price.

When there is a profit, the Selling Price is a percentage of the Cost Price given by:

$SP_{new} = CP \times (1 + \frac{\text{Profit}\%}{100})$

Substitute the calculated CP and the desired Profit Percentage:

$SP_{new} = \textsf{₹}20,000 \times (1 + \frac{8}{100})$

$SP_{new} = \textsf{₹}20,000 \times (1 + 0.08)$

$SP_{new} = \textsf{₹}20,000 \times 1.08$

Calculate the new selling price:

$SP_{new} = 20000 \times 1.08 = 21600$

$SP_{new} = \textsf{₹}21,600$

To gain $8\%$, the shopkeeper should sell the sofa set for $\textsf{₹}21,600$.

He should sell it for $\textsf{₹}21,600$ to gain $8\%$.

Given:

Simple Interest (SI) = $\textsf{₹}2,800$

Time (T) = $4$ years

Rate of Interest (R) = $7\%$ per annum

To Find:

The Principal (P) or the sum of money.

Solution:

The formula for calculating Simple Interest is:

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

We are given SI, R, and T, and we need to find P. We can rearrange the formula to solve for P:

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

Substitute the given values into the formula:

Calculate the denominator:

$7 \times 4 = 28$

So, equation (i) becomes:

$P = \frac{\textsf{₹}2800 \times 100}{28}$

$P = \frac{\textsf{₹}280000}{28}$

Perform the division:

$P = \textsf{₹}\frac{\cancel{2800}^{100} \times 100}{\cancel{28}_{1}}$

$P = \textsf{₹}100 \times 100$

$P = \textsf{₹}10,000$

The principal amount is $\textsf{₹}10,000$.

The sum of money that will yield a simple interest of $\textsf{₹}2,800$ in $4$ years at $7\%$ per annum is $\textsf{₹}10,000$.

Given:

Total number of chairs bought = $20$

Cost price per chair = $\textsf{₹}800$

Number of chairs sold at first price = $16$

Selling price per chair (for first batch) = $\textsf{₹}1000$

Number of remaining chairs = $20 - 16 = 4$

Selling price per chair (for remaining batch) = $\textsf{₹}700$

To Find:

1. The total profit or loss.

2. The profit or loss percentage.

Solution:

First, calculate the total Cost Price (CP) of all the chairs.

Total CP = Number of chairs $\times$ Cost price per chair

Total CP $= 20 \times \textsf{₹}800$

Total CP $= \textsf{₹}16,000$

Next, calculate the total Selling Price (SP) of all the chairs.

The chairs were sold in two batches:

SP of the first batch (16 chairs) = Number of chairs $\times$ Selling price per chair

SP of first batch $= 16 \times \textsf{₹}1000$

SP of first batch $= \textsf{₹}16,000$

Number of remaining chairs $= 20 - 16 = 4$

SP of the remaining batch (4 chairs) = Number of chairs $\times$ Selling price per chair

SP of remaining batch $= 4 \times \textsf{₹}700$

SP of remaining batch $= \textsf{₹}2,800$

Total Selling Price (SP) = SP of first batch + SP of remaining batch

Total SP $= \textsf{₹}16,000 + \textsf{₹}2,800$

Total SP $= \textsf{₹}18,800$

Now, compare the Total Selling Price (SP) and the Total Cost Price (CP) to determine profit or loss.

Total CP $= \textsf{₹}16,000$

Total SP $= \textsf{₹}18,800$

Since Total SP ($\textsf{₹}18,800$) is greater than Total CP ($\textsf{₹}16,000$), there is a profit.

Calculate the amount of profit:

Profit = Total Selling Price (SP) - Total Cost Price (CP)

Profit $= \textsf{₹}18,800 - \textsf{₹}16,000$

Profit $= \textsf{₹}2,800$

Now, calculate the profit percentage. The profit percentage is calculated on the Total Cost Price.

Profit Percentage $= \frac{\text{Profit}}{\text{Total Cost Price (CP)}} \times 100\%$

Substitute the values of Profit and Total CP:

Profit Percentage $= \frac{\textsf{₹}2,800}{\textsf{₹}16,000} \times 100\%$

Profit Percentage $= \frac{2800}{16000} \times 100\%$

Simplify the fraction:

Profit Percentage $= \frac{\cancel{2800}^{28}}{\cancel{16000}_{160}} \times 100\%$

Profit Percentage $= \frac{28}{160} \times 100\%$

Divide numerator and denominator by their GCD, which is $4$:

Profit Percentage $= \frac{\cancel{28}^{7}}{\cancel{160}_{40}} \times 100\%$

Profit Percentage $= \frac{7}{40} \times 100\%$

Profit Percentage $= \frac{7}{\cancel{40}_{2}} \times \cancel{100}^{5}\%$

Profit Percentage $= \frac{7 \times 5}{2}\%$

Profit Percentage $= \frac{35}{2}\%$

Profit Percentage $= 17.5\%$

The shopkeeper made a total profit of $\textsf{₹}2,800$, and the profit percentage is $17.5\%$.

The total profit is $\textsf{₹}2,800$.

The profit percentage is $17.5\%$.

Given:

Total number of students in the class = $50$

Percentage of boys in the class = $60\%$

Percentage of boys who passed = $80\%$ (of the boys)

Percentage of girls who passed = $90\%$ (of the girls)

To Find:

1. The number of boys who passed the exam.

2. The number of girls who passed the exam.

3. The total percentage of students who passed the exam.

Solution:

First, find the number of boys and girls in the class.

Number of boys $= 60\%$ of Total students

Number of boys $= \frac{60}{100} \times 50$

Number of boys $= \frac{60}{\cancel{100}_{2}} \times \cancel{50}^{1}$

Number of boys $= \frac{60}{2} = 30$

So, there are $30$ boys in the class.

Number of girls = Total students - Number of boys

Number of girls $= 50 - 30 = 20$

Alternatively, the percentage of girls is $100\% - 60\% = 40\%$.

Number of girls $= 40\%$ of $50$

Number of girls $= \frac{40}{100} \times 50 = \frac{40}{2} = 20$

So, there are $20$ girls in the class.

Now, find the number of boys who passed.

Number of boys who passed $= 80\%$ of the number of boys

Number of boys who passed $= 80\%$ of $30$

Number of boys who passed $= \frac{80}{100} \times 30$

Number of boys who passed $= \frac{80}{\cancel{100}_{10}} \times \cancel{30}^{3}$

Number of boys who passed $= \frac{\cancel{80}^{8}}{\cancel{10}_{1}} \times 3$

Number of boys who passed $= 8 \times 3 = 24$

Next, find the number of girls who passed.

Number of girls who passed $= 90\%$ of the number of girls

Number of girls who passed $= 90\%$ of $20$

Number of girls who passed $= \frac{90}{100} \times 20$

Number of girls who passed $= \frac{90}{\cancel{100}_{5}} \times \cancel{20}^{1}$

Number of girls who passed $= \frac{90}{5} = 18$

Now, find the total number of students who passed.

Total students who passed = Number of boys who passed + Number of girls who passed

Total students who passed $= 24 + 18 = 42$

Finally, find the total percentage of students who passed the exam. This is calculated based on the total number of students in the class.

Total percentage of students who passed $= \frac{\text{Total students who passed}}{\text{Total number of students}} \times 100\%$

Total percentage of students who passed $= \frac{42}{50} \times 100\%$

Simplify the expression:

Total percentage of students who passed $= \frac{42}{\cancel{50}_{1}} \times \cancel{100}^{2}\%$

Total percentage of students who passed $= 42 \times 2\%$

Total percentage of students who passed $= 84\%$

The number of boys who passed is $24$.

The number of girls who passed is $18$.

The total percentage of students who passed the exam is $84\%$.

Given:

Principal (P) = $\textsf{₹}15,000$

Rate of Interest (R) = $10\%$ per annum

Time (T) = $3$ years

To Find:

(a) Simple Interest (SI) after $3$ years.

(b) Total Amount (A) to be paid back after $3$ years.

(c) The amount still due if $\textsf{₹}18,000$ is paid back.

Solution:

We will address each part of the question separately.

(a) Find the simple interest after 3 years.

We use the formula for Simple Interest:

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

Substitute the given values:

$SI = \frac{15000 \times 10 \times 3}{100}$

Calculate the value:

$SI = \frac{15000 \times 30}{100}$

$SI = \frac{450000}{100}$

$SI = 4500$

The simple interest after $3$ years is $\textsf{₹}4500$.

(b) Find the amount to be paid back after 3 years.

The total amount to be paid back is the sum of the Principal and the Simple Interest.

Amount (A) = Principal (P) + Simple Interest (SI)

Substitute the values of Principal and the calculated Simple Interest:

$A = \textsf{₹}15000 + \textsf{₹}4500$

$A = \textsf{₹}19500$

The total amount to be paid back after $3$ years is $\textsf{₹}19500$.

(c) If the borrower pays back ₹18,000 at the end of 3 years, how much more is due?

The total amount due is the amount calculated in part (b), which is $\textsf{₹}19500$.

Amount paid back = $\textsf{₹}18,000$

Amount still due = Total Amount Due - Amount Paid Back

Substitute the values:

Amount still due $= \textsf{₹}19500 - \textsf{₹}18000$

Amount still due $= \textsf{₹}1500$

If the borrower pays back $\textsf{₹}18,000$, an amount of $\textsf{₹}1500$ is still due.

Summary of results:

(a) The simple interest after $3$ years is $\textsf{₹}4500$.

(b) The total amount to be paid back after $3$ years is $\textsf{₹}19500$.

(c) The amount still due is $\textsf{₹}1500$.

Given:

Ram's income is $20\%$ more than Shyam's income.

To Find:

The percentage by which Shyam's income is less than Ram's income.

Solution:

Let's assume Shyam's income is $\textsf{₹}100$ for simplicity.

Shyam's income $= \textsf{₹}100$

Ram's income is $20\%$ more than Shyam's income.

Increase in income for Ram $= 20\%$ of Shyam's income

Increase $= 20\%$ of $\textsf{₹}100$

Increase $= \frac{20}{100} \times \textsf{₹}100 = \textsf{₹}20$

Ram's income = Shyam's income + Increase

Ram's income $= \textsf{₹}100 + \textsf{₹}20 = \textsf{₹}120$

Now we need to find by what percentage Shyam's income is less than Ram's income.

Difference between Ram's income and Shyam's income $= \text{Ram's income} - \text{Shyam's income}$

Difference $= \textsf{₹}120 - \textsf{₹}100 = \textsf{₹}20$

We need to express this difference as a percentage of Ram's income.

Percentage less $= \frac{\text{Difference}}{\text{Ram's income}} \times 100\%$

Percentage less $= \frac{\textsf{₹}20}{\textsf{₹}120} \times 100\%$

Percentage less $= \frac{20}{120} \times 100\%$

Simplify the fraction:

Percentage less $= \frac{\cancel{20}^{1}}{\cancel{120}_{6}} \times 100\%$

Percentage less $= \frac{1}{6} \times 100\%$

Percentage less $= \frac{100}{6}\%$

Percentage less $= \frac{50}{3}\%$

Percentage less $= 16\frac{2}{3}\%$ or approximately $16.67\%$

Alternatively, let Shyam's income be $S$.

Ram's income $R = S + 20\% \text{ of } S = S + \frac{20}{100} S = S + 0.2S = 1.2S$

The difference between Ram's income and Shyam's income is $R - S = 1.2S - S = 0.2S$.

The percentage by which Shyam's income is less than Ram's income is $\frac{R - S}{R} \times 100\%$.

Percentage less $= \frac{0.2S}{1.2S} \times 100\%$

Percentage less $= \frac{0.2}{1.2} \times 100\%$

Percentage less $= \frac{2}{12} \times 100\%$

Percentage less $= \frac{1}{6} \times 100\% = 16\frac{2}{3}\%$.

Shyam's income is $16\frac{2}{3}\%$ less than Ram's income.

Shyam's income is $16\frac{2}{3}\%$ less than Ram's income.