Chapter 1 Fractions in Disguise (Class 8 - Latest Maths NCERT (Ganita Prakash II) Solutions)

To Find:

Conversion of given fractions into percentages.

Solution:

In the Indian school system, a common method to convert a fraction into a percentage is to multiply the fraction by $100$ and add the percent symbol ($\%$).

(i) $\frac{3}{5}$

$\text{Percentage} = 3 \times 20 = 60\%$

(ii) $\frac{7}{14}$

$\text{Percentage} = \frac{1}{\cancel{2}_1} \times \cancel{100}^{50} = 50\%$

(iii) $\frac{9}{20}$

$\text{Percentage} = \frac{9}{\cancel{20}_1} \times \cancel{100}^{5}$

$\text{Percentage} = 9 \times 5 = 45\%$

(iv) $\frac{72}{150}$

$\text{Percentage} = \frac{72}{\cancel{150}_3} \times \cancel{100}^{2}$

$\text{Percentage} = \frac{\cancel{72}^{24}}{3} \times 2$

$\text{Percentage} = 24 \times 2 = 48\%$

(v) $\frac{1}{3}$

$\text{Percentage} = \frac{1}{3} \times 100 = \frac{100}{3}\%$

On dividing $100$ by $3$, we get a recurring decimal:

$\text{Percentage} = 33.33\dots\% = 33\frac{1}{3}\%$

(vi) $\frac{5}{11}$

$\text{Percentage} = \frac{5}{11} \times 100 = \frac{500}{11}\%$

By long division:

$\text{Percentage} \approx 45.45\%$

Summary Table:

Given:

Total number of marbles with Nandini = $25$

Number of white marbles = $15$

To Find:

Percentage of white marbles.

Solution:

The percentage is calculated by the formula:

$\text{Percentage} = \frac{\text{Value}}{\text{Total Value}} \times 100$

Substituting the values:

We can simplify the fraction $\frac{15}{25}$ by dividing both the numerator and the denominator by their HCF, which is $5$.

$\text{Percentage} = \frac{\cancel{15}^3}{\cancel{25}_5} \times 100$

$\text{Percentage} = \frac{3}{\cancel{5}_1} \times \cancel{100}^{20}$

$\text{Percentage} = 3 \times 20 = 60\%$

Conclusion:

Nandini has $60\%$ white marbles. Therefore, the correct option is (iv).

Alternate Solution:

In Indian perspective, we can use the unitary method. If there were $100$ marbles, the number of white marbles would be four times as much as there are for $25$.

Since $25 \times 4 = 100$,

Then $15 \times 4 = 60$.

Thus, the percentage is $60\%$.

Given:

Total number of students in the school = $80$

Number of students who come by walking = $15$

To Find:

The percentage of students who walk to school.

Solution:

To find the percentage, we express the number of walking students as a fraction of the total students and then multiply by $100$.

Substituting the values:

$\text{Percentage} = \frac{15}{80} \times 100$

Simplifying the fraction by cancelling common factors:

$\text{Percentage} = \frac{\cancel{15}^{3}}{\cancel{80}_{16}} \times 100$

$\text{Percentage} = \frac{3 \times \cancel{100}^{25}}{\cancel{16}_{4}}$

$\text{Percentage} = \frac{3 \times 25}{4} = \frac{75}{4}$

$\text{Percentage} = 18.75\%$

Indian Perspective:

In Indian villages and small towns, a large number of students walk to school daily. Understanding this percentage helps school authorities in Pradhan Mantri Gram Sadak Yojana planning or providing bicycle schemes like those in many Indian states.

Alternate Solution:

We can use the unitary method. If $80$ students correspond to $100\%$, then $1$ student corresponds to $\frac{100}{80}\%$.

Therefore, $15$ students correspond to:

$15 \times \frac{100}{80} = 15 \times 1.25 = 18.75\%$

Solution:

Since the specific image is not visible here, we follow the general Indian curriculum logic for "matching based on visual estimation". We divide the race track into logical halves and quarters ($25\%, 50\%, 75\%$).

Steps for Estimation:

1. If a runner is at the start, they are at $0\%$.

2. If they are exactly in the middle of the track, they have completed $50\%$.

3. If they are midway between the start and the middle, they have completed $25\%$.

4. If they are midway between the middle and the finish line, they have completed $75\%$.

Typical Matching Table (Example):

Conclusion:

To solve this, look at the relative distance each friend has covered from the starting point to the finish line and select the percentage value that matches their visual position on the track.

Solution:

We will compare these using logical reasoning and benchmarks ($0, \frac{1}{2}, 1$).

(i) $50\%$ ____ $5\%$

$50\%$ represents half of a whole, whereas $5\%$ is a very small fraction (one-twentieth).

Result: $50\% > 5\%$

(ii) $\frac{5}{10}$ ____ $50\%$

$\frac{5}{10}$ simplified is $\frac{1}{2}$. We know that $50\%$ also represents $\frac{1}{2}$ of the total.

Result: $\frac{5}{10} = 50\%$

(iii) $\frac{3}{11}$ _____ $61\%$

$\frac{3}{11}$ is less than $\frac{3}{6}$ (which would be $50\%$). Since $3$ is much less than half of $11$, this fraction is clearly less than $50\%$. On the other hand, $61\%$ is greater than $50\%$.

Result: $\frac{3}{11} < 61\%$

(iv) $30\%$ ____ $\frac{1}{3}$

We know that $\frac{1}{3}$ is approximately $33.33\%$. Comparing $30\%$ with $33.33\%$, it is evident that $30\%$ is smaller.

Result: $30\% < \frac{1}{3}$

Summary:

1. (i) $>$

2. (ii) $=$

3. (iii) $<$

4. (iv) $<$

Solution:

The bar models help us visualize percentages as parts of a whole. In the Indian School Context, this is often called the "Unitary Method" approach applied to visual segments.

(i) Worked out example:

Total value = $75$. The bar is divided into $5$ equal parts.

Value of $1$ part = $\frac{75}{5} = 15$.

Percentage of $1$ part = $\frac{100\%}{5} = 20\%$.

The arrow indicates $4$ parts, which is $4 \times 15 = 60$. Thus, $60$ is $80\%$ of $75$.

(ii) Orange Bar Model:

Given: Total value = $90$. The bar is divided into $10$ equal parts.

To Find: Percentage of one block and the value indicated by the arrow ($7$ blocks).

Step 1: Calculate the percentage of one block.

Step 2: Calculate the value of one block.

Step 3: Find the missing value indicated by the arrow ($7$ blocks).

Missing numbers for (ii): The percentage of one block is $10\%$ and the value for the arrow is $63$.

(iii) Green Bar Model:

Given: Total value = $140$. The bar is divided into $4$ equal parts.

To Find: Percentage of one block and the total value indicated by $3$ blocks.

Step 1: Percentage of one block.

Step 2: Value of one block.

Value = $\frac{140}{4} = 35$.

Step 3: Value of $3$ blocks (indicated by the arrow).

Missing numbers for (iii): The percentage of one block is $25\%$ and the indicated value is $105$.

Solution:

(i) $25\%$ of $160$

Calculation: $\frac{25}{100} \times 160 = \frac{1}{4} \times 160 = 40$.

Bar Model: A bar representing $160$ divided into $4$ parts (each $25\%$), where one part is shaded as $40$.

(ii) $16\%$ of $250$

Calculation: $\frac{16}{100} \times 250 = \frac{16 \times \cancel{250}^{5}}{\cancel{100}_{2}} = 8 \times 5 = 40$.

Bar Model: A bar representing $250$, where a small segment representing $16\%$ is marked as $40$.

(iii) $62\%$ of $360$

Calculation: $\frac{62}{100} \times 360 = 0.62 \times 360 = 223.2$.

(iv) $140\%$ of $40$

Calculation: $\frac{140}{100} \times 40 = 1.4 \times 40 = 56$.

Note: Here the result is greater than the original number because the percentage is over $100\%$.

(v) $1\%$ of $1$ hour

Indian Perspective: We usually convert time to minutes or seconds for smaller percentages.

$1$ hour = $60$ minutes = $3600$ seconds.

Value = $\frac{1}{100} \times 3600 \text{ seconds} = 36 \text{ seconds}$.

(vi) $7\%$ of $10$ kg

$10$ kg = $10,000$ grams.

Value = $\frac{7}{100} \times 10 \text{ kg} = 0.7 \text{ kg} = 700 \text{ g}$.

Summary of results:

Given:

Total volume of deep orange paint = $60$ ml.

Fraction of red paint used = $\frac{3}{4}$.

To Find:

The volume of red paint in ml.

Solution:

The volume of red paint can be found by multiplying the total volume by the given fraction.

Simplifying the calculation:

Alternate Solution:

We can convert the fraction to a percentage: $\frac{3}{4} = 75\%$.

$75\%$ of $60$ ml = $0.75 \times 60 = 45$ ml.

Conclusion: Surya used $45$ ml of red paint to make the mixture.

Solution:

(i) $50\%$ of $510$ < $50\%$ of $515$

Since the percentage is the same ($50\%$), the expression with the larger base number will have the larger value. $515 > 510$, hence the second quantity is larger.

(ii) $37\%$ of $148$ < $73\%$ of $148$

Since the base number is the same ($148$), the expression with the higher percentage will be larger. $73\% > 37\%$.

(iii) $29\%$ of $43$ < $92\%$ of $110$

In the second quantity, both the percentage ($92\%$) and the base number ($110$) are significantly larger than those in the first quantity. Estimation clearly shows the second value is much higher.

(iv) $30\%$ of $40$ < $40\%$ of $50$

Let's verify by calculation:

Since $12 < 20$, the symbol is '$<$'.

(v) $45\%$ of $200$ > $10\%$ of $490$

Estimation: $45\%$ of $200$ is slightly less than half, which is around $90$. $10\%$ of $490$ is $49$. Clearly, $90 > 49$.

(vi) $30\%$ of $80$ > $24\%$ of $64$

Calculation: $30\%$ of $80$ is $24$. For the second part, $24\%$ of $100$ is $24$, so $24\%$ of $64$ must be much smaller than $24$.

Indian Perspective:

Estimation is a skill frequently used by Indian shopkeepers and housewives to quickly calculate discounts or budget household expenses without needing a calculator.

Solution:

(i) Given $30\%$ of $k = 70$:

To find $60\%$, we double the value: $70 \times 2 = 140$.

To find $90\%$, we triple the value: $70 \times 3 = 210$.

To find $120\%$, we quadruple the value: $70 \times 4 = 280$.

Ans: 140, 210, 280

(ii) Given $100\%$ of $m = 215$:

To find $10\%$, divide by $10$: $215 \div 10 = 21.5$.

To find $1\%$, divide by $100$: $215 \div 100 = 2.15$.

To find $6\%$, multiply $1\%$ by $6$: $2.15 \times 6 = 12.9$.

Ans: 21.5, 2.15, 12.9

(iii) Given $90\%$ of $n = 270$:

To find $9\%$, divide by $10$: $270 \div 10 = 27$.

To find $18\%$, multiply $9\%$ value by $2$: $27 \times 2 = 54$.

To find $100\%$, we find $10\%$ first ($30$) and multiply by $10$: $300$.

Ans: 27, 54, 300

(iv) Self-made questions:

1. $20\%$ of $x$ is $50$, find $10\%$, $40\%$, and $100\%$.

2. $100\%$ of $y$ is $400$, find $5\%$, $25\%$, and $150\%$.

Solution:

(i) $3$ is $x\%$ of $300$

Ans: 1

(ii) $x$ is $40\%$ of $4$

Ans: 1.6

(iii) $40$ is $80\%$ of $x$

Ans: 50

Summary:

(i) 1; (ii) 1.6; (iii) 50.

Given:

Quantity A: $10\%$ of a day

Quantity B: $1\%$ of a week

To Find:

Which duration is longer.

Solution:

First, let us convert both quantities into the same unit (hours) to compare them accurately, as is common in Indian mathematics problems.

Step 1: Calculating $10\%$ of a day

Converting $0.4$ hours to minutes: $0.4 \times 60 = 24 \text{ minutes}$. So, $10\%$ of a day is $2 \text{ hours } 24 \text{ minutes}$.

Step 2: Calculating $1\%$ of a week

Converting $0.68$ hours to minutes: $0.68 \times 60 \approx 40.8 \text{ minutes}$. So, $1\%$ of a week is approx $1 \text{ hour } 41 \text{ minutes}$.

Comparison:

Since $2.4 \text{ hours} > 1.68 \text{ hours}$, $10\%$ of a day is longer than $1\%$ of a week.

Challenge Questions for Peers:

1. Is $25\%$ of a $\textsf{₹} 1000$ note more than $10\%$ of a $\textsf{₹} 2000$ note and a $\textsf{₹} 500$ note combined?

2. Is $50\%$ of an hour longer than $5\%$ of a full day?

Solution:

We need to represent the ratio of fodder eaten to fodder given as a percentage for different days.

Observation:

By analyzing the data, we observe the following patterns:

1. Increasing Trend: As the days progress, the percentage of fodder eaten by the bull is steadily increasing.

2. Gap Reduction: Although the amount of fodder left uneaten is always constant ($1$ unit), it represents a smaller and smaller proportion of the total as the total quantity grows.

3. Approaching 100%: The percentage values are approaching $100\%$ but will never quite reach it as long as the bull leaves $1$ unit. This is a great practical example of limits in mathematics, often discussed in higher secondary schools in India.

Given:

Portion of plantation completed = $20\%$

Time taken = $18$ days

To Find:

Time taken to complete $100\%$ (the entire plantation).

Solution:

We can use the Unitary Method, which is the standard technique taught in Indian arithmetic.

Time taken for $20\%$ work = $18$ days

Alternate Solution:

Since $20\%$ is $\frac{1}{5}$ of the whole plantation, the total time will be $5$ times the time taken for $20\%$.

Total time = $18 \times 5 = 90$ days.

Importance of the Assumption:

The assumption that the rate of work stays the same is necessary because:

1. Environmental Factors: In Indian coffee estates (like those in Coorg or Wayanad), heavy rains or heatwaves can slow down work. If it rains, workers cannot pick berries at the same speed.

2. Labor Consistency: It assumes the number of workers remains the same and they don't get tired or fall ill over the long period of $90$ days.

3. Terrain: Some parts of the plantation might be on steeper slopes than others, making the work harder in those sections.

Without this assumption, we cannot use simple Direct Proportion to solve the problem.

Given:

Total duration of training = $90$ minutes

Percentage for Warm up = $10\%$

Percentage for Play = $80\%$

Percentage for Cool down = $10\%$

To Find:

The time duration (in minutes) for each specific activity.

Solution:

We calculate the time for each activity by finding the respective percentage of the total $90$ minutes.

1. Duration for Warm up:

$\text{Time} = \frac{1}{\cancel{10}} \times 9\cancel{0} = 9 \text{ minutes}$

2. Duration for Play:

$\text{Time} = \frac{8}{\cancel{10}} \times 9\cancel{0} = 8 \times 9 = 72 \text{ minutes}$

3. Duration for Cool down:

$\text{Time} = 9 \text{ minutes}$

Verification:

Total time = $9 + 72 + 9 = 90$ minutes. This matches the given total duration.

Indian Perspective:

Badminton has seen a massive surge in popularity in India due to the success of stars like P.V. Sindhu, Saina Nehwal, and Lakshya Sen. Professional academies across the country follow strict scientific training schedules where warm-up and cool-down are essential to prevent injuries.

Given:

Percentage of world population in Northern Hemisphere = $90\%$

Approximate current world population (2024-25) $\approx 8.2$ billion

To Find:

Approximate number of people living in the Northern Hemisphere.

Solution:

We need to find $90\%$ of $8.2$ billion.

$\text{Population} = 0.9 \times 8.2$

$\text{Population} = 7.38$ billion

Indian Perspective:

India is located entirely in the Northern Hemisphere and is currently the most populous country in the world. India's massive population of over $1.4$ billion contributes significantly to this $90\%$ statistic.

Final Answer:

Approximately $7.38$ billion people live in the Northern Hemisphere.

Given:

Proportion for $4$ people: Rava ($40\%$), Sugar ($40\%$), Ghee ($20\%$).

(i) Solution for 8 people:

When we increase the quantity of a dish to serve more people, the proportion (percentage) of each ingredient remains exactly the same to maintain the taste. Only the actual weight/volume of the ingredients increases.

Proportions for 8 people:

Rava: $40\%$

Sugar: $40\%$

Ghee: $20\%$

(ii) Solution for total weight of $2$ kg:

Total weight = $2$ kg = $2000$ grams.

1. Quantity of Rava:

$\text{Rava} = 40\% \text{ of } 2000 \text{ g}$

$\text{Rava} = \frac{40}{100} \times 2000 = 800 \text{ g}$

2. Quantity of Sugar:

$\text{Sugar} = 40\% \text{ of } 2000 \text{ g}$

$\text{Sugar} = \frac{40}{100} \times 2000 = 800 \text{ g}$

3. Quantity of Ghee:

$\text{Ghee} = 20\% \text{ of } 2000 \text{ g}$

$\text{Ghee} = \frac{20}{100} \times 2000 = 400 \text{ g}$

Verification:

Total weight = $800 + 800 + 400 = 2000$ g = $2$ kg.

Indian Perspective:

Halwa is a traditional Indian sweet prepared during festivals and celebrations. Whether it is Sooji ka Halwa or Gajar ka Halwa, Ghee (clarified butter) is a vital ingredient in Indian kitchens, often used in a specific ratio to ensure the right texture and aroma.

Given:

Cost Price (CP) of the geometry box = $\textsf{₹} 75$

Selling Price (SP) of the geometry box = $\textsf{₹} 110$

To Find:

Profit margin with respect to the cost (Profit Percentage).

Solution:

First, we need to calculate the actual profit earned by the shopkeeper.

Now, we calculate the profit margin with respect to the cost price:

$\text{Profit Margin} = \frac{35}{75} \times 100$

Simplifying the fraction:

$\text{Profit Margin} = \frac{\cancel{35}^7}{\cancel{75}_{15}} \times 100$

$\text{Profit Margin} = \frac{7 \times \cancel{100}^{20}}{\cancel{15}_3} = \frac{140}{3}$

$\text{Profit Margin} \approx 46.67\%$

Indian Perspective:

In Indian retail markets, shopkeepers often aim for such margins on stationery items like geometry boxes to cover their overhead expenses and local transportation costs.

Given:

Cost of materials (CP) = $\textsf{₹} 475$

Desired Profit Margin = $50\%$

To Find:

Selling Price (SP) of the chair.

Solution:

First, we calculate the amount of profit the carpenter wants to earn.

$\text{Profit} = \frac{50}{100} \times 475 = \frac{1}{2} \times 475 = \textsf{₹} 237.50$

Now, we find the Selling Price:

$\text{SP} = 475 + 237.50$

$\text{SP} = \textsf{₹} 712.50$

Conclusion:

The carpenter should sell each chair for $\textsf{₹} 712.50$ to achieve a $50\%$ profit margin.

Alternate Solution:

We can use the direct formula for Selling Price:

$\text{SP} = \text{CP} \times \left( \frac{100 + \text{Gain}\%}{100} \right)$

$\text{SP} = 475 \times \frac{150}{100} = 475 \times 1.5 = \textsf{₹} 712.50$

Given:

Total Sales (Revenue/SP) = $\textsf{₹} 2.5$ crore

Profit Margin = $25\%$ (on cost)

To Find:

Total Expenditure (Cost Price).

Solution:

Let the total expenditure be $x$.

According to the problem, the sales figure includes the cost plus a $25\%$ profit on that cost.

$2.5 = x + 0.25x$

$2.5 = 1.25x$

Solving for $x$:

$x = 2$

Conclusion:

The total expenditure of the company was $\textsf{₹} 2$ crore. The profit earned was $\textsf{₹} 0.5$ crore.

Indian Perspective:

In India, business revenue is often expressed in terms like "Crores" or "Lakhs". A $25\%$ profit margin is considered very strong for manufacturing or service sectors in the Indian economy.

Given:

Original Price (Marked Price) = $\textsf{₹} 300$

Discount Percentage = $25\%$

To Find:

Final Price Anwar has to pay.

Solution:

First, we calculate the discount amount in rupees.

$\text{Discount} = \frac{25}{100} \times 300 = \frac{1}{4} \times 300 = \textsf{₹} 75$

Now, we find the final price after subtracting the discount from the original price.

$\text{Final Price} = \textsf{₹} 225$

Conclusion:

Anwar will have to pay $\textsf{₹} 225$ to buy the shirt.

Alternate Solution:

Since the discount is $25\%$, Anwar only pays $(100\% - 25\%) = 75\%$ of the original price.

$\text{Final Price} = 75\% \text{ of } 300 = \frac{75}{100} \times 300 = 75 \times 3 = \textsf{₹} 225$.

Given:

Price in 2015 (Original Price) = $\textsf{₹} 60$

Price in 2025 (New Price) = $\textsf{₹} 100$

To Find:

Percentage increase in the price of petrol.

Solution:

First, we find the absolute increase in price:

Now, we calculate the percentage increase using the formula:

Substituting the values from equation (i):

Rounding off, we get $66.66\%$.

Conclusion:

The correct option is (iv) $66.66\%$.

Indian Perspective:

In India, petrol prices are a major economic indicator. A significant increase over a decade affects the Common Man (Aam Aadmi) by increasing the cost of logistics, transportation, and daily commuting via two-wheelers and public transport.

Given:

Selling Price (after discount) = $\textsf{₹} 4,40,000$

Discount Percentage = $15\%$

To Find:

Original Price (Marked Price) of the car.

Solution:

Let the original price of the car be $x$.

If the discount is $15\%$, then the selling price is $100\% - 15\% = 85\%$ of the original price.

Simplifying the fraction by dividing by $5$:

Conclusion:

The original price of the car was approximately $\textsf{₹} 5,17,647.06$.

Alternate Solution:

We can use the formula: $\text{Marked Price} = \frac{\text{Selling Price} \times 100}{100 - \text{Discount}\%}$.

$\text{MP} = \frac{440000 \times 100}{100 - 15} = \frac{440000 \times 100}{85} = \textsf{₹} 5,17,647.06$.

Part 1: Percentage of votes for the winner

Given:

Total votes = $1600$

Winner's votes = $500$

Solution:

Part 2: Guessing the minimum number of candidates

Solution:

The winner got $31.25\%$ of the votes. In a standard election (like the Lok Sabha elections in India), a winner is the person with the highest number of votes (First-Past-The-Post system).

1. If there were only 2 candidates: The winner must have more than $50\%$ of the votes. Since $31.25\% < 50\%$, there must be more than 2 candidates.

2. If there were 3 candidates: The remaining $100\% - 31.25\% = 68.75\%$ would be shared by 2 others. The average for the others would be $68.75 \div 2 = 34.375\%$. If one of them got $34.375\%$, the person with $31.25\%$ would not be the winner.

3. If there were 4 candidates: The remaining $68.75\%$ is shared by 3 others. The average for others would be $68.75 \div 3 \approx 22.92\%$. Since $31.25\% > 22.92\%$, it is possible for the person with $31.25\%$ to be the winner.

Conclusion:

The winner got $31.25\%$ of the total votes. The minimum number of candidates who stood for the election must be 4.

Given:

Price of rice in $2024$ (Original Price) = $\textsf{₹} 38$

Price of rice in $2025$ (New Price) = $\textsf{₹} 42$

To Find:

Rate of inflation (Percentage increase in price).

Solution:

First, we calculate the increase in the price of rice:

Now, we find the percentage increase using the formula:

$\text{Inflation Rate} = \frac{\text{Increase}}{\text{Original Price}} \times 100$

Indian Perspective:

In India, rice is a staple food for a majority of the population. An inflation rate of over $10\%$ in basic food items like rice is a significant concern for middle-class and low-income families, as it directly impacts their monthly kitchen budget and overall cost of living.

Given:

Percentage increase = $20\%$

Resulting number after increase = $90$

To Find:

The original number.

Solution:

Let the original number be $x$.

According to the problem, the original number plus $20\%$ of itself equals $90$.

To simplify, multiply the numerator and denominator by $10$:

Conclusion:

The original number is 75.

Alternate Solution:

If a number is increased by $20\%$, it becomes $120\%$ of its original value.

Therefore, $120\% \text{ of } x = 90$.

$\frac{120}{100} \times x = 90 \Rightarrow x = \frac{90 \times 100}{120} = \frac{3 \times 100}{4} = 75$.

Given:

Selling Price (SP) of each buffalo = $\textsf{₹} 80,000$

Buffalo 1: Profit = $5\%$

Buffalo 2: Loss = $10\%$

To Find:

Overall profit or loss amount.

Solution:

First, we calculate the Cost Price (CP) for each buffalo separately.

For Buffalo 1 (Profit $5\%$):

$\text{CP}_1 = \frac{\text{SP} \times 100}{100 + \text{Profit}\%}$

$\text{CP}_1 = \frac{1600000}{21} \approx \textsf{₹} 76,190.48$

For Buffalo 2 (Loss $10\%$):

$\text{CP}_2 = \frac{\text{SP} \times 100}{100 - \text{Loss}\%}$

$\text{CP}_2 = \frac{800000}{9} \approx \textsf{₹} 88,888.89$

Calculation of Overall Result:

Total Selling Price (Total SP) = $80,000 + 80,000 = \textsf{₹} 1,60,000$

Total Cost Price (Total CP) = $76,190.48 + 88,888.89 = \textsf{₹} 1,65,079.37$

Since Total CP > Total SP, the milkman has incurred an overall loss.

Indian Perspective:

In many parts of Rural India, the sale and purchase of livestock like buffaloes is a primary source of income for milkmen (Doodhwalas). Calculating exact profits and losses helps these small-scale entrepreneurs manage their dairy business effectively.

Given:

Population of elephants in the last decade = $p$

Percentage increase in population = $5\%$

To Find:

The current population of elephants.

Solution:

An increase of $5\%$ means the current population is the sum of the original population and the additional $5\%$ growth.

Now, the current population is found by adding this increase to the original population:

Conclusion:

The correct option is (iv) $p \times 1.05$.

Indian Perspective:

Elephant conservation is a critical part of Indian wildlife policy. In national parks like Kaziranga, Jim Corbett, or Periyar, population growth is tracked every decade to ensure the survival of the Asiatic Elephant, which is revered in Indian culture as a symbol of wisdom and strength (often associated with Lord Ganesha).

Given:

Percentage fall in demand = $85\%$

Solution:

When we say a quantity has "fallen by $85\%$", it means we subtract $85\%$ from the original $100\%$ value.

This means the current demand is $15\%$ of what it used to be a decade ago.

Evaluation of Statements:

1. Statement (i) says it is $85\%$, which is incorrect because $85\%$ is the part that was lost, not the part that remains.

2. Statement (iii) says "The demand now is $15\%$ of the demand a decade ago," which matches our calculation.

Conclusion:

The statement that means the same is (iii).

Indian Perspective:

This scenario is highly relevant to the Indian consumer market. Over the last decade, with the arrival of high-quality smartphones in India, the demand for standalone digital cameras (like point-and-shoot cameras) has seen a massive decline of approximately $85\%$ or more, as most Indians now prefer using their mobile phones for photography and social media.

Given:

Principal ($P$) = $\textsf{₹} 20,000$

Rate of Interest ($R$) = $10\%$ p.a.

Time ($T$ or $n$) = $2$ years

To Find:

1. Amount without compounding (Simple Interest).

2. Amount with annual compounding (Compound Interest).

3. Comparison between the two.

Solution:

Case 1: Without Compounding (Simple Interest)

In Indian banking, simple interest is often used for short-term loans or specific savings schemes like some post office deposits.

$SI = \frac{20000 \times 10 \times 2}{100}$

$SI = 200 \times 10 \times 2 = \textsf{₹} 4,000$

Total Amount ($A_{SI}$) = $P + SI$

Case 2: With Annual Compounding (Compound Interest)

Most Indian commercial banks (like SBI or HDFC) use compound interest for Fixed Deposits (FDs).

$A_{CI} = 20000 \left(1 + \frac{10}{100}\right)^2$

$A_{CI} = 20000 \left(1.1\right)^2$

Comparison:

Difference = $A_{CI} - A_{SI}$

Difference = $24200 - 24000 = \textsf{₹} 200$

One gets $\textsf{₹} 200$ more if the interest is compounded annually.

Given:

Principal ($P$) = $\textsf{₹} 20,000$

Rate of Interest ($R$) = $5\%$ p.a.

Time ($T$ or $n$) = $4$ years

Solution:

Case 1: Without Compounding (Simple Interest)

$SI = 200 \times 20 = \textsf{₹} 4,000$

Total Amount ($A_{SI}$) = $20000 + 4000 = \textsf{₹} 24,000$

Case 2: With Annual Compounding (Compound Interest)

$A_{CI} = 20000 \times (1.05)^4$

$A_{CI} = 20000 \times 1.21550625$

Comparison:

Difference = $24310.13 - 24000 = \textsf{₹} 310.13$

In this case, one gets $\textsf{₹} 310.13$ more through compounding.

Observations and Discussion:

1. Equality of Simple Interest: Interestingly, in both Question 1 and Question 2, the Simple Interest earned is exactly the same ($\textsf{₹} 4,000$). This is because the product of Rate and Time is the same ($10 \times 2 = 20$ and $5 \times 4 = 20$).

2. Power of Time in Compounding: Even though the product of $R \times T$ is constant, the Compound Interest is higher in the second case ($\textsf{₹} 4,310.13$) compared to the first ($\textsf{₹} 4,200$). This shows that Time ($n$) has a more powerful effect in the compound interest formula because it sits in the exponent position.

3. The Compounding Effect: We observe that $A_{CI}$ is always greater than $A_{SI}$ for any period longer than one year. In the Indian financial context, this is why financial advisors suggest starting investments early; even with a lower interest rate, a longer duration significantly boosts the final corpus due to the "magic of compounding."

Summary Table:

Given:

Invested Amount (Principal) = $p$

Time period ($T$) = $4$ years

Rate of Interest ($R$) = $6\%$ p.a.

Interest type: Not compounded (Simple Interest).

To Find:

The correct expression for the total amount after $4$ years.

Solution:

In Indian arithmetic, when interest is not compounded, we use the Simple Interest formula. The interest is calculated as:

The fraction $\frac{6}{100}$ can be written in decimal form as $0.06$. Substituting this in equation (i):

The Total Amount Jasmine will receive is the sum of the original principal and the interest earned:

Substituting the values from equation (ii):

Conclusion:

Comparing equation (iii) with the given options, the correct expression is (vii).

Given:

Principal Amount ($P$) = $\textsf{₹} 50,000$

Rate of Interest ($R$) = $7\%$ p.a.

Time period ($T$ or $n$) = $3$ years

To Find:

1. Simple Interest (without compounding).

2. The difference between Compound Interest and Simple Interest.

Solution:

Step 1: Calculating Interest without Compounding (Simple Interest)

Post offices in India are very popular for such safe investment schemes. The simple interest is:

Step 2: Calculating Interest with Annual Compounding

First, we find the total amount ($A$) using the compound interest formula:

Now, we find the Compound Interest ($CI$):

Step 3: Calculating the Difference

Conclusion:

One would get $\textsf{₹} 10,500$ interest without compounding. With compounding, one would get $\textsf{₹} 752.15$ more.

Given:

For both: Principal ($P$) = $\textsf{₹} 12,500$, Time ($n$ or $T$) = $3$ years.

Giridhar: Rate ($R_1$) = $12\%$ (Simple Interest).

Raghava: Rate ($R_2$) = $10\%$ (Compound Interest).

To Find:

Who pays more interest and the difference between their interests.

Solution:

Step 1: Interest paid by Giridhar (Simple Interest)

$I_G = \frac{12500 \times 12 \times 3}{100}$

$I_G = 125 \times 36 = \textsf{₹} 4,500$.

Step 2: Interest paid by Raghava (Compound Interest)

First, we calculate the Amount ($A$):

$A = 12500 \left(1 + \frac{10}{100}\right)^3$

$A = 12500 \times (1.1)^3$

$A = 12500 \times 1.331 = \textsf{₹} 16,637.50$.

Now, Compound Interest ($I_R$):

$I_R = 16637.50 - 12500 = \textsf{₹} 4,137.50$.

Step 3: Comparison

Comparing $I_G = \textsf{₹} 4,500$ and $I_R = \textsf{₹} 4,137.50$.

Difference = $4500 - 4137.50 = \textsf{₹} 362.50$.

Conclusion: Giridhar pays more interest by $\textsf{₹} 362.50$.

Indian Perspective:

In Indian rural economies, loans for agriculture or small businesses (like those from NABARD or local cooperatives) often vary between simple and compound interest. It is vital for borrowers to understand that a higher rate of Simple Interest can sometimes be costlier than a lower rate of Compound Interest over a short period.

Given:

Principal ($P$) = $\textsf{₹} 1,000$, Rate ($R$) = $10\%$, Target Amount ($A$) = $\textsf{₹} 2,000$ (Double).

Solution:

Case 1: Without Compounding (Linear Growth)

Interest required = $2000 - 1000 = \textsf{₹} 1,000$.

$1000 = 100 \times T \Rightarrow T = 10 \text{ years}$.

Case 2: With Compounding (Exponential Growth)

$2 = (1.1)^n$.

Using the Rule of 72 (common in Indian finance): $n \approx 72 / 10 = 7.2$ years.

By calculation: $(1.1)^7 \approx 1.948$ and $(1.1)^8 \approx 2.143$. Thus, it takes approximately $7.27$ years.

Observations:

1. Simple Interest (Linear): The amount grows by a fixed quantity ($\textsf{₹} 100$) every year. The graph is a Straight Line.

2. Compound Interest (Exponential): The interest itself earns interest. The growth accelerates over time. The graph is a Curve.

Conclusion: Compounding is indeed exponential growth, while non-compounding is linear growth.

Given:

Current Population ($P$) = $1.5$ crore.

Growth Rate ($R$) = $3\%$ p.a., Time ($n$) = $3$ years.

Solution:

Population growth follows the Compound Interest formula as the increase happens on the updated population of each year.

$A = 1.5 \left(1 + \frac{3}{100}\right)^3$

$A = 1.5 \times (1.03)^3$

$A = 1.5 \times 1.092727 = 1.6390905 \text{ crore}$.

Rounding to two decimal places, the population is approximately $1.64$ crore.

Indian Perspective:

Urbanization in Indian metros like Bengaluru, Pune, or Gurugram often sees rapid population growth. Estimating this helps the Municipal Corporations plan for infrastructure, water supply, and housing (like PMAY schemes).

Given:

Initial count ($P$) = $5,06,000$.

Growth rate ($R$) = $2.5\%$ per hour, Time ($n$) = $2$ hours.

Solution:

The increase in bacteria is a biological process that follows the compounding rule.

$A = 506000 \times (1.025)^2$

$A = 506000 \times 1.050625$

$A = 531616.25$.

Since the number of bacteria must be a whole number, we round it to $5,31,616$.

Final Result:

The number of bacteria after 2 hours is $5,31,616$.

Given:

Population of Bengaluru in the year $2000 = 50$ lakhs

Percentage growth/comparison for $2025 = 250\%$ of the population in $2000$

To Find:

The population of Bengaluru in the year $2025$.

Solution:

To find the current population, we calculate $250\%$ of the base population from $2000$.

Indian Perspective:

Bengaluru, known as the Silicon Valley of India, has seen an unprecedented population explosion due to the IT boom. A jump from $50$ lakhs to $1.25$ crore ($125$ lakhs) reflects the massive urbanization and migration of professionals from across the country to Karnataka's capital.

Given:

Total World Population $\approx 8.2$ billion (or $8,200$ million)

Germany $\approx 83$ million

India $\approx 1.46$ billion (or $1,460$ million)

Bangladesh $\approx 175$ million

USA $\approx 347$ million

Solution:

We calculate the percentage share of each country using the formula:

$\text{Percentage Share} = \frac{\text{Country Population}}{\text{World Population}} \times 100$

1. India:

2. USA:

3. Bangladesh:

4. Germany:

Matching Results:

Indian Perspective:

India is now the most populous country in the world, surpassing China. Representing nearly one-fifth of the global population, India's demographics play a central role in global economic and environmental discussions.

Given:

Base price of mobile phone = $\textsf{₹} 8,250$

GST rate = $18\%$

Solution:

The final price is the base price plus the GST amount calculated on that base price.

$\text{Final Price} = \text{Base Price} + (18\% \text{ of Base Price})$

Alternatively, we can factor out $8250$:

Conclusion:

The correct expressions are (v) $8250 \times 1.18$ and (vi) $8250 + 8250 \times 0.18$.

Indian Perspective:

The Goods and Services Tax (GST) was introduced in India in 2017 to create a unified tax structure. For electronic gadgets like mobile phones, the standard GST rate is often $18\%$, which is why bills at shops like Croma or Reliance Digital show this specific tax breakdown.

Given:

Initial population of mice = $p$.

Month 1 change = $+5\%$.

Month 2 change = $-2\%$.

Month 3 change = $-3\%$.

To Find:

The correct mathematical expression for the population after three months and its comparison with the initial population.

Solution:

Percentage changes are compounded because each month's change is calculated on the population resulting from the previous month.

1. After Month 1, population becomes $p \times (1 + 0.05) = p \times 1.05$.

2. After Month 2, population becomes $(p \times 1.05) \times (1 - 0.02) = p \times 1.05 \times 0.98$.

3. After Month 3, population becomes $(p \times 1.05 \times 0.98) \times (1 - 0.03) = p \times 1.05 \times 0.98 \times 0.97$.

Now, let us calculate the product of the multipliers:

$1.05 \times 0.98 \times 0.97 = 0.99813$.

Since $0.99813$ is less than 1, the final population $0.99813p$ is less than the initial population $p$.

Conclusion:

The true statements are (ii) and (vi).

Indian Perspective:

In Indian agriculture and biology studies, such population trends are analyzed to understand the growth or decline of pests or livestock. Understanding that successive percentage gains and losses do not simply cancel each other out is a crucial skill in financial literacy and data analysis across the country.

Given:

Initial profit margin = $35\%$ above cost price.

Discount offered = $30\%$ on the selling price.

To Find:

Whether the shopkeeper makes a profit or a loss.

Solution:

Let us assume the Cost Price (CP) of the product is $\textsf{₹} 100$.

Step 1: Calculate the initial Selling Price (SP).

$\text{Profit margin} = 35\%$.

Step 2: Calculate the discount and new Selling Price.

A $30\%$ discount is offered on the SP ($\textsf{₹} 135$).

$\text{Discount amount} = 30\% \text{ of } 135$

$\text{Discount amount} = \frac{30}{100} \times 135 = 0.3 \times 135 = \textsf{₹} 40.5$

Step 3: Compare CP and new SP.

CP = $\textsf{₹} 100$ and New SP = $\textsf{₹} 94.5$.

Since $\text{CP} > \text{SP}$, the shopkeeper makes a loss.

Conclusion:

The shopkeeper makes a loss of $5.5\%$. The reason is that the $30\%$ discount is calculated on a larger base value (the increased selling price), which results in a reduction that is greater than the original $35\%$ increase on the cost price.

Indian Perspective:

During festive sales like Diwali or Dussehra in India, many shopkeepers use strategies of "markup and then discount." This mathematical exercise helps customers realize that a $30\%$ discount on a $35\%$ markup does not mean they are still making a $5\%$ profit.

Given:

A square grid divided into different regions A, B, C, D, and E.

To Find:

The percentage of the total area occupied by region ‘E’.

Solution:

Let us count the units based on the dot intervals in the grid.

1. The total grid size is $8$ units $\times$ $8$ units (based on $9$ dots horizontally and vertically).

2. Region ‘E’ is a right-angled triangle.

Its base is $4$ units (from the middle vertical line to the left edge).

Its height is $4$ units (from the bottom horizontal line to the middle horizontal divider).

3. Calculate the area of region ‘E’:

4. Calculate the percentage of area occupied by E:

Conclusion:

Region ‘E’ occupies $12.5\%$ of the total area.

Alternate Solution:

Region D and E together form a square of $4 \times 4 = 16$ units. Since the line divides this square exactly in half, E is half of that area.

$\text{Area of D+E} = 16 \text{ units}$.

$\text{Area of E} = 8 \text{ units}$.

$\text{Fraction} = \frac{8}{64} = \frac{1}{8}$, which is known to be $12.5\%$ in Indian school mathematics benchmarks.

Solution:

Let us calculate each pair to identify the pattern:

Case 1:

$5\%$ of $40 = \frac{5}{100} \times 40 = 2$

$40\%$ of $5 = \frac{40}{100} \times 5 = 2$

Case 2:

$25\%$ of $12 = \frac{25}{100} \times 12 = \frac{1}{4} \times 12 = 3$

$12\%$ of $25 = \frac{12}{100} \times 25 = \frac{12}{4} = 3$

Case 3:

$15\%$ of $60 = \frac{15}{100} \times 60 = 9$

$60\%$ of $15 = \frac{60}{100} \times 15 = 9$

Observation:

In all the cases above, the results of the two calculations are identical. This means $x\%$ of $y$ is always equal to $y\%$ of $x$.

Algebraic Justification:

Let us consider two numbers $x$ and $y$.

According to the Commutative Property of Multiplication, which is a fundamental concept in Indian Vedic Mathematics as well as modern algebra, we know that:

Therefore, from equations (i) and (ii), we can conclude that:

Indian Perspective:

This "Percent Switch" trick is extremely useful in Indian competitive exams like the CAT or Bank PO. It is often easier to calculate $12\%$ of $50$ (which is $50\%$ of $12 = 6$) than to do the math the other way around.

Given:

Grade 8 students = $40\%$ of total students.

Girls in Grade 8 = $60\%$ of Grade 8 students.

Solution (i):

To find the percentage of Grade 8 girls relative to the entire group, we multiply the two percentages.

So, $24\%$ of the total students are Grade 8 girls.

Solution (ii):

Total number of students = $160$.

Number of Grade 8 girls = $24\%$ of $160$.

Note: In a practical Indian school scenario, the number of students must be a whole number. This suggests that the total count or percentages in the problem are approximations. Mathematically, the result is $38.4$.

Indian Perspective:

School excursions, often called 'Educational Tours' or 'Shala Pravas' in India, are a mandatory part of the curriculum in many states. This data helps teachers manage logistics, such as the number of seats to reserve for girls in the Bharat Benz buses commonly used for such trips.

Given:

Selling Price (SP) of $3$ pencils = Cost Price (CP) of $5$ pencils.

To Find:

Profit or Loss percentage.

Solution:

Let the Cost Price (CP) of $1$ pencil be $\textsf{₹} x$.

As per the question:

Now, we find the Selling Price of $1$ pencil:

Since $\frac{5}{3}x$ (approx $1.67x$) is greater than $x$, the shopkeeper makes a profit.

Now, calculate the Profit Percentage:

Alternate Solution:

Let CP of $1$ pencil = $\textsf{₹} 3$ (HCF of 3 and 5 for easy calculation).

CP of $5$ pencils = $5 \times 3 = \textsf{₹} 15$.

SP of $3$ pencils = $\textsf{₹} 15$. So, SP of $1$ pencil = $15/3 = \textsf{₹} 5$.

Profit per pencil = $5 - 3 = \textsf{₹} 2$.

Profit $\%$ = $(2/3) \times 100 = 66.67\%$.

Given:

Increase in Year 1 = $3\%$.

Increase in Year 2 = $4\%$.

Solution:

This is a case of Successive Percentage Increase. Let the initial bus fare be $\textsf{₹} 100$.

Step 1: After the first year increase ($3\%$)

New Fare = $100 + (3\% \text{ of } 100) = \textsf{₹} 103$.

Step 2: After the second year increase ($4\%$)

The $4\%$ increase is applied to the new fare of $\textsf{₹} 103$.

Step 3: Calculate total percentage increase

Overall Percentage Increase = $7.12\%$.

Indian Perspective:

Bus fares for State Road Transport Corporations (like MSRTC, UPSRTC, or KSRTC) in India are usually revised every year based on fuel price fluctuations. It is important for commuters to realize that a $3\%$ and $4\%$ increase results in slightly more than $7\%$ due to the compounding effect.

Given:

Percentage increase in length = $10\%$

The area of the rectangle remains unchanged.

To Find:

The percentage decrease in the breadth of the rectangle.

Solution:

Let the initial length be $L$ and the initial breadth be $B$.

Initial Area ($A$) = $L \times B$

New Length ($L'$) = $L + 10\% \text{ of } L = 1.1L$

Let the new breadth be $B'$. Since the area remains the same:

Dividing both sides by $L$:

Now, we find the decrease in breadth:

$\text{Decrease} = B - B' = B - \frac{10}{11}B = \frac{1}{11}B$

Percentage decrease:

$\text{Percentage} \approx 9.09\%$

Indian Perspective:

This type of problem is very common in Indian scholarship exams and the National Talent Search Examination (NTSE). It teaches students that changes in dimensions are inversely proportional when the product (area) is constant.

Alternate Solution:

Using the shortcut formula for constant product:

If one factor increases by $x\%$, the other must decrease by $\left(\frac{x}{100+x} \times 100\right)\%$.

$\text{Decrease} = \frac{10}{100+10} \times 100 = \frac{10}{110} \times 100 = \frac{100}{11} = 9.09\%$.

Given:

Total weight of the chips packet = $65$ g

Proportions: Potato ($70\%$), Vegetable oil ($24\%$), Salt ($3\%$), and Spices ($3\%$).

To Find:

The weight of each individual ingredient in grams.

Solution:

We find the weight of each ingredient by calculating its percentage of the total $65$ g weight.

1. Weight of Potato:

$\text{Weight} = 45.5 \text{ g}$

2. Weight of Vegetable Oil:

$\text{Weight} = 15.6 \text{ g}$

3. Weight of Salt:

$\text{Weight} = 1.95 \text{ g}$

4. Weight of Spices:

Summary of Ingredients:

Indian Perspective:

In India, the Food Safety and Standards Authority of India (FSSAI) mandates that all packaged food must display nutritional information and ingredient lists. Checking these percentages helps health-conscious Indian consumers manage their intake of salt and fats.

(i) Solution: Effective Price per Item

Assume the price of one item is $\textsf{₹} 100$.

Shop A (Buy 1 get 1): You pay for $1$ item but get $2$ items. Total cost = $\textsf{₹} 100$.

$\text{Effective price} = \frac{100}{2} = \textsf{₹} 50$.

Shop B (Buy 2 get 1): You pay for $2$ items but get $3$ items. Total cost = $\textsf{₹} 200$.

$\text{Effective price} = \frac{200}{3} \approx \textsf{₹} 66.67$.

Shop C (Buy 3 get 1): You pay for $3$ items but get $4$ items. Total cost = $\textsf{₹} 300$.

$\text{Effective price} = \frac{300}{4} = \textsf{₹} 75$.

Order (Cheapest to Costliest): Shop A < Shop B < Shop C.

(ii) Solution: Percentage Discount

$\text{Discount } \% = \frac{\text{Free Items}}{\text{Total Items}} \times 100$

Shop A: $\frac{1}{2} \times 100 = 50\%$

Shop B: $\frac{1}{3} \times 100 = 33.33\%$

Shop C: $\frac{1}{4} \times 100 = 25\%$

(iii) Choice for 4 Items:

If we need $4$ items, let's calculate the total cost in each shop:

Shop A: We "Buy 2 and get 2 free". Total cost = $2 \times 100 = \textsf{₹} 200$.

Shop B: We "Buy 2 and get 1 free" (3 items) and then buy $1$ more. Total cost = $200 + 100 = \textsf{₹} 300$.

Shop C: We "Buy 3 and get 1 free". Total cost = $3 \times 100 = \textsf{₹} 300$.

Decision: I would choose Shop A because it offers the lowest total cost of $\textsf{₹} 200$ for 4 items.

Indian Perspective:

These "BOGO" (Buy One Get One) deals are very popular in Indian malls during the End of Season Sale (EOSS). Understanding the "effective discount" helps smart Indian shoppers save money and avoid falling for marketing gimmicks where the discount seems larger than it actually is.

Given:

To Find:

The number of left-handed people who must leave to make the new left-handed percentage $98\%$.

Solution:

First, we determine the number of left-handed and right-handed people initially.

Number of left-handed people = $99\%$ of $100 = 99$.

Let the number of left-handed people who leave be $x$.

New total number of people = $100 - x$.

New number of left-handed people = $99 - x$.

According to the question, the new percentage of left-handed people should be $98\%$.

Cross-multiplying the terms:

$100(99 - x) = 98(100 - x)$

$9900 - 100x = 9800 - 98x$

$9900 - 9800 = 100x - 98x$

Verification:

If $50$ left-handed people leave, the new total is $100 - 50 = 50$ people. The remaining left-handers are $99 - 50 = 49$.

Percentage = $\frac{49}{50} \times 100 = 98\%$. (Correct)

Alternate Solution:

We know the number of right-handed people stays exactly $1$. In the final situation, these $1$ right-handed person must represent $2\%$ of the total ($100\% - 98\% = 2\%$).

If $2\%$ of Total = $1$, then $100\%$ of Total = $\frac{1}{2} \times 100 = 50$.

To reach a total of $50$ from $100$, exactly $50$ people must leave.

Conclusion:

$50$ left-handed people have to leave the room.

Solution:

This graph is based on the National Sample Survey (NSS) Round 79, which provides vital data for Digital India planning. Let us evaluate each statement based on the provided percentages:

(i) Valid: Looking at the bars, the combined percentages for 'Twenties' (Male: $37\%$, Female: $26\%$) are visibly higher than any other group. Teenagers are the second highest ($29\%$ and $24\%$).

(ii) Valid: In every single age category (Children, Teenage, Twenties, etc.), the dark blue bar (Female) is shorter than the orange bar (Male). This indicates a gender gap in digital literacy across all ages in 2023.

(iii) Invalid: The graph represents the percentage of people within that age group who can use a computer. It does not provide the absolute population count of how many teenagers or twenty-year-olds live in the country.

(iv) Invalid: For people in their thirties, the female percentage is $14\%$ and the male is $25\%$. A "quarter" means $25\%$. Since one gender is exactly $25\%$ and the other is much lower, the average for the whole group is less than a quarter.

(v) Valid: "1 in 10" is equal to $10\%$. For seniors, the female percentage is $2\%$ and the male is $4\%$. Both are significantly less than $10\%$.

(vi) Invalid: "Half" means $50\%$. Even in the most literate group (Twenties), the highest bar is $37\%$, which is much lower than $50\%$.

Conclusion:

The valid statements are (i), (ii), and (v).

Indian Perspective:

Data like this from the Ministry of Statistics and Programme Implementation (MOSPI) helps the Government of India identify where to focus initiatives like the Pradhan Mantri Gramin Digital Saksharta Abhiyan (PMGDISHA) to bridge the digital divide, especially among women and senior citizens.