GSEB Class 7 Maths Solutions Chapter 8 Comparing Quantities InText Questions

Exam Tip: When dealing with fractions that don't easily convert to a denominator of 100, remember that \( \frac{1}{3} = 33.\overline{3}\% \) and \( \frac{2}{3} = 66.\overline{6}\% \). Always ensure the sum of percentages equals 100%.

Think, Discuss and Write (Page 160)

Question 1. Look at the examples below and in each of them, discuss which is better for comparison.
Answer: (i) In the atmosphere, 1 g of air contains:
.78 g Nitrogen
.21% Oxygen
.01 g Other gas
or
78% Nitrogen
21 g Oxygen
1% Other Gas
(ii) A shirt has:
\( \frac { 3 }{ 5 } \) Cotton
\( \frac { 2 }{ 5 } \) Polyster
or
60% Cotton
40% Polyster
The second option (i.e. percentage) is always better for comparison in both these situations. Percentages make it easier to understand relative amounts quickly. They provide a clear standard for comparison, regardless of the original quantity.
In simple words: When comparing things, using percentages (like 78% Nitrogen or 60% Cotton) is usually simpler and clearer than using grams or fractions. Percentages help us easily see the proportional amounts.

Exam Tip: Percentages normalize values to a common base (100), making them highly effective for comparing different quantities or proportions, especially in real-world contexts.

Think, Discuss and Write (Page 161)

Question 1. (i) Can you eat 50% of a cake? Can you eat 100% of a cake? Can you eat 150% of a cake? (ii) Can a price of an item go up by 50%? Can a price of an item go up by 100%? Can a price of an item go up by 150%?
Answer:
(i) Yes, we can eat 50% of a cake. This means eating half of it.
Yes, we can eat 100% of a cake. This means eating the whole cake.
No, we cannot eat 150% of a cake. You cannot eat more than the entire cake itself.
(ii) Yes, a price of an item can go up by 50%. This means it becomes 1.5 times the original price.
Yes, a price of an item can go up by 100%. This means it doubles in price.
Yes, a price of an item can go up by 150%. This means it becomes 2.5 times the original price.
In simple words: You can only eat up to 100% of a cake because that's all there is. But a price can go up by more than 100% because it's a value, not a fixed physical amount.

Exam Tip: Differentiate between quantities that have a fixed maximum (like a whole cake, which is 100%) and quantities that can increase indefinitely (like prices or growth, which can exceed 100%).

Try These (Page 161)

Question 1. Convert the following to per cents:
(a) \( \frac { 12 }{ 16 } \)
(b) 3.5
(c) \( \frac { 49 }{ 50 } \)
(d) \( \frac { 2 }{ 2 } \)
(e) 0.05
Answer:
(a) We have: \( \frac { 12 }{ 16 } = \frac { 12 }{ 16 } \times 100\% = (3 \times 25)\% = 75\% \)
(b) We have: \( 3.5 = 3.5 \times \frac { 100 }{ 100 } = (3.5 \times 100)\% = 350\% \)
(c) We have: \( \frac { 49 }{ 50 } = \frac { 49 }{ 50 } \times 100\% = (49 \times 2)\% = 98\% \)
(d) We have: \( \frac { 2 }{ 2 } = 1 = (1 \times 100)\% = 100\% \)
(e) We have: \( 0.05 = 0.05 \times \frac { 100 }{ 100 } = (0.05 \times 100)\% = 5\% \)
In simple words: To change a fraction or a decimal into a percentage, you just multiply it by 100. For example, 3.5 becomes 350%.

Exam Tip: Remember that "per cent" means "out of one hundred." To convert any number (fraction, decimal, or whole number) into a percentage, multiply it by 100%.

Question 2.
(i) Out of 32 students, 8 are absent. What per cent of the students are absent?
(ii) There are 25 radios, 16 of them are out of order. What per cent of radios are out of order?
(iii) A shop has 500 parts, out of which 5 are defective. What per cent are defective?
(iv) There are 120 voters, 90 of them voted yes. What per cent voted yes?
Answer:
(i) The fraction of absent students is \( \frac { 8 }{ 32 } \).
So, \( \frac { 8 }{ 32 } \times 100\% = (8 \times 3.125)\% = 25\% \).
\( \implies \) Therefore, 25% of the students are absent.
(ii) The fraction of radios out of order is \( \frac { 16 }{ 25 } \).
So, \( \frac { 16 }{ 25 } \times 100\% = (16 \times 4)\% = 64\% \).
\( \implies \) Therefore, 64% of the radios are out of order.
(iii) The fraction of defective parts is \( \frac { 5 }{ 500 } \).
So, \( \frac { 5 }{ 500 } \times 100\% = (5 \times 0.2)\% = 1\% \).
\( \implies \) Therefore, 1% of the parts are defective.
(iv) The fraction of voters who voted yes is \( \frac { 90 }{ 120 } \).
So, \( \frac { 90 }{ 120 } \times 100\% = (0.75 \times 100)\% = 75\% \).
\( \implies \) Therefore, 75% of the voters voted 'Yes'.
In simple words: For each part, we find what fraction the specified group makes up (like absent students or broken radios). Then, we multiply that fraction by 100 to get the percentage for that group.

Exam Tip: When calculating percentages, clearly identify the part and the whole. The formula is \( \frac{\text{Part}}{\text{Whole}} \times 100\% \). Simplification before multiplication can make calculations easier.

Try These (Page 162)

Question 1. Fill in the blanks:
(i) 35% + ______ % = 100%
(ii) 64% + 20% + ______ % = 100%
(iii) 45% = 100% - ______ %
(iv) 70% = ______ % - 30%
Answer:
(i) To find the missing percentage, we subtract 35 from 100: \( 100 - 35 = 65 \).
\( \implies \) So, 35% + 65% = 100%.
(ii) First, add the given percentages: \( 64 + 20 = 84 \). Then subtract this sum from 100: \( 100 - 84 = 16 \).
\( \implies \) So, 64% + 20% + 16% = 100%.
(iii) To find the missing percentage, subtract 45 from 100: \( 100 - 45 = 55 \).
\( \implies \) So, 45% = 100% - 55%.
(iv) To find the missing percentage, add 70 and 30: \( 70 + 30 = 100 \).
\( \implies \) So, 70% = 100% - 30%.
In simple words: These problems are about making percentages add up to 100% or finding a part of a whole. We use addition and subtraction to figure out the missing numbers.

Exam Tip: Remember that total percentage always sums to 100%. Use simple arithmetic to find the missing parts of a whole or in an equation.

Question 2. If 65% of students in a class have a bicycle, what per cent of the students do not have bicycles?
Answer: The total percentage of students is always 100%.
If 65% of students possess bicycles, then the remaining portion of students do not have them.
Remaining part of students \( = 100\% - 65\% = (100 - 65)\% = 35\% \).
Thus, 35% of students do not own bicycles.
In simple words: If 65 out of every 100 students have a bicycle, then the rest (35 out of 100) do not.

Exam Tip: When a problem states a percentage of a group, the remaining percentage (to make 100%) represents the other part of the group. Always start with 100% as the total.

Question 3. We have a basket full of apples, oranges and mangoes. If 50% are apples, 30% are oranges, then what per cent are mangoes?
Answer: In the basket, the quantity of apples is 50%, and the quantity of oranges is 30%.
The total percentage of fruits in the basket is 100%.
The quantity of mangoes in the basket \( = 100\% - (50\% + 30\%) \).
\( = [100 - (50 + 30)]\% \)
\( = [100 - 80]\% = 20\% \).
Thus, the percentage of mangoes in the basket is 20%.
In simple words: If apples and oranges make up 80% of the basket, then mangoes must be the remaining 20% to reach a total of 100% of fruits.

Exam Tip: For problems involving proportions of a whole, add up the given percentages and subtract from 100% to find the unknown percentage.

Think, Discuss and Write (Page 162)

Question 1. Consider the expenditure made on a dress 20% on embroidery, 50% on cloth, 30% on stitching. Can you think of more such examples. Do it yourself.
Answer: This question asks you to think of your own examples where percentages are used to show parts of a total. You should work on this yourself to explore different scenarios.
In simple words: Think about everyday things where different parts add up to a whole, like how much time you spend on homework, chores, and playing, or how a budget is split.

Exam Tip: When providing examples, ensure the percentages logically sum to 100% of the total quantity or budget being described.

Try These (Page 163)

Question 1. What per cent of these figures are shaded?
(i)
(ii)
Answer:
(i) For the first figure, the total number of parts is 4. The number of shaded parts is 3.
\( \implies \) Percentage of the shaded parts \( = \frac { 3 }{ 4 } \times 100\% \)
\( = (3 \times 25)\% \)
\( = 75\% \).

(ii) For the tangram figure, the solution states the shaded parts are \( \frac { 1 }{ 4 }, \frac { 1 }{ 8 } \text{ and } \frac { 1 }{ 8 } \).
The total shaded parts \( = \frac { 1 }{ 4 } + \frac { 1 }{ 8 } + \frac { 1 }{ 8 } \).
To add these fractions, we find a common denominator, which is 8.
\( = \frac { 2 }{ 8 } + \frac { 1 }{ 8 } + \frac { 1 }{ 8 } = \frac { 2+1+1 }{ 8 } = \frac { 4 }{ 8 } = \frac { 1 }{ 2 } \).
\( \implies \) Percentage of the shaded parts \( = \frac { 1 }{ 2 } \times 100\% = 50\% \).

In simple words: For the first figure, three out of four parts are shaded, which means 75% is shaded. For the second tangram figure, if we add up the fractions of the shaded parts (one-fourth, one-eighth, and one-eighth), they equal half of the total, meaning 50% is shaded.

Exam Tip: When dealing with shaded figures, first determine the total number of equal parts and the number of shaded parts. Then, express this as a fraction and convert it to a percentage.

Try These (Page 164)

Question 1. Find:
(a) 50% of 164
(b) 75% of 12
(c) \( 12\frac { 1 }{ 2 }\% \) of 64
Answer:
(a) To find 50% of 164, we convert 50% to a fraction or decimal: \( 50\% = \frac { 50 }{ 100 } = \frac { 1 }{ 2 } \).
So, 50% of 164 \( = \frac { 1 }{ 2 } \times 164 = 82 \).
Thus, 50% of 164 is 82.
(b) To find 75% of 12, we convert 75% to a fraction or decimal: \( 75\% = \frac { 75 }{ 100 } = \frac { 3 }{ 4 } \).
So, 75% of 12 \( = \frac { 3 }{ 4 } \times 12 = 3 \times 3 = 9 \).
Thus, 75% of 12 is 9.
(c) To find \( 12\frac { 1 }{ 2 }\% \) of 64, we first convert the mixed percentage to an improper fraction: \( 12\frac { 1 }{ 2 } = \frac { 25 }{ 2 } \).
So, \( 12\frac { 1 }{ 2 }\% = \frac { 25 }{ 2 } \times \frac { 1 }{ 100 } = \frac { 25 }{ 200 } = \frac { 1 }{ 8 } \).
Then, \( 12\frac { 1 }{ 2 }\% \) of 64 \( = \frac { 1 }{ 8 } \times 64 = 8 \).
Thus, \( 12\frac { 1 }{ 2 }\% \) of 64 is 8.
In simple words: To find a percentage of a number, change the percentage into a fraction (like 50% becomes 1/2) or a decimal (like 75% becomes 0.75), then multiply that fraction or decimal by the number.

Exam Tip: Always convert percentages to their fractional or decimal equivalents before performing calculations. Remember that \( 12\frac { 1 }{ 2 }\% \) is a common fraction equivalent to \( \frac{1}{8} \).

Question 2. 8% children of a class of 25 like getting wet in the rain. How many children like getting wet in the rain?
Answer: To find how many children like getting wet, we calculate 8% of the total number of children, which is 25.
\( 8\% \text{ of } 25 = \frac { 8 }{ 100 } \times 25 \).
\( = \frac { 8 }{ 4 } = 2 \).
\( \implies \) Therefore, 2 children enjoy getting wet in the rain.
In simple words: To find how many children like getting wet, we calculate 8 percent of the 25 students. This calculation shows that 2 children enjoy the rain.

Exam Tip: When finding a percentage of a whole number, express the percentage as a fraction (e.g., \( \frac{8}{100} \)) and then multiply by the given number.

Try These (Page 164)

Question 1. 9 is 25% of what number?
Answer: Let the desired number be \( x \).
We are told that 25% of \( x \) is equal to 9.
So, we can write the equation as: \( 25\% \times x = 9 \).
Converting 25% to a fraction: \( \frac { 25 }{ 100 } \times x = 9 \).
To solve for \( x \), we multiply both sides by \( \frac { 100 }{ 25 } \):
\( x = 9 \times \frac { 100 }{ 25 } \).
\( x = 9 \times 4 \).
\( x = 36 \).
Thus, 9 represents 25% of 36.
In simple words: If 9 is one-quarter (25%) of a number, we multiply 9 by 4 to find the full number, which is 36.

Exam Tip: When asked to find the whole number given a percentage of it, set up an equation where the percentage (as a decimal or fraction) times the unknown number equals the given part. Solve for the unknown.

Question 2. 75% of what number is 15?
Answer: Let the required number be \( x \).
We know that 75% of \( x \) is 15.
So, we write the equation: \( 75\% \times x = 15 \).
Converting 75% to a fraction: \( \frac { 75 }{ 100 } \times x = 15 \).
To find \( x \), we multiply both sides by \( \frac { 100 }{ 75 } \):
\( x = 15 \times \frac { 100 }{ 75 } \).
\( x = \frac { 1500 }{ 75 } \).
\( x = 20 \).
Thus, 75% of 20 is 15.
In simple words: If 15 is 75 percent (or three-quarters) of a number, we can find the whole number by dividing 15 by 3, then multiplying by 4, which gives us 20.

Exam Tip: You can quickly solve these by realizing that \( 75\% = \frac{3}{4} \). If \( \frac{3}{4} \) of a number is 15, then \( \frac{1}{4} \) is 5, and the whole number is \( 4 \times 5 = 20 \).

Try These (Page 166)

Question 1. Divide 15 sweets between Manu and Sonu so that they get 20% and 80% of them.
Answer: We need to distribute 15 sweets such that Manu receives 20% and Sonu receives 80%.
Share of Manu \( = 20\% \text{ of } 15 \text{ sweets} \).
\( = \frac { 20 }{ 100 } \times 15 \).
\( = \frac { 1 }{ 5 } \times 15 = 3 \text{ sweets} \).
Share of Sonu \( = 80\% \text{ of } 15 \text{ sweets} \).
\( = \frac { 80 }{ 100 } \times 15 \).
\( = \frac { 4 }{ 5 } \times 15 = 4 \times 3 = 12 \text{ sweets} \).
Thus, Manu gets 3 sweets and Sonu gets 12 sweets.
In simple words: To share 15 sweets, Manu gets 20% (which is 3 sweets), and Sonu gets 80% (which is 12 sweets).

Exam Tip: When dividing a quantity by percentage, calculate each share separately. Always double-check that the sum of the individual shares equals the total quantity.

Question 2. If angles of a triangle are in the ratio 2:3:4. Find the value of each angle.
Answer: We know that the total sum of the three angles in any triangle is always 180°.
The given ratios of the angles are 2:3:4.
First, find the sum of these ratio parts: \( 2 + 3 + 4 = 9 \).
Now, find the value of each angle by dividing 180° by the sum of the ratios and multiplying by each ratio part:
First angle: \( \frac { 2 }{ 9 } \times 180° = 2 \times 20° = 40° \).
Second angle: \( \frac { 3 }{ 9 } \times 180° = 3 \times 20° = 60° \).
Third angle: \( \frac { 4 }{ 9 } \times 180° = 4 \times 20° = 80° \).
Thus, the angles are 40°, 60°, and 80°.
In simple words: The angles of a triangle add up to 180 degrees. If their ratio is 2:3:4, we divide 180 by the total ratio (9) to find what one "part" is worth (20 degrees). Then, we multiply 20 by each ratio number to get the individual angle sizes.

Exam Tip: Always remember that the sum of angles in a triangle is 180°. For ratio problems, calculate the total ratio parts and divide the total quantity by this sum to find the value of one ratio part.

Try These (Page 167)

Question 1. Find percentage of increase or decrease:
(i) Price of shirt decreased from Rs 80 to Rs 60.
(ii) Marks in a test increased from 20 to 30.
Answer:
(i) Initial price of the shirt = Rs 80.
Decreased price of the shirt = Rs 60.
The decrease in price \( = \text{Rs } 80 - \text{Rs } 60 = \text{Rs } 20 \).
Percentage decrease of price \( = \frac { \text{Decrease in Price} }{ \text{Initial Price} } \times 100\% \).
\( = \frac { 20 }{ 80 } \times 100\% \)
\( = \frac { 1 }{ 4 } \times 100\% = 25\% \).
(ii) Initial marks = 20.
Increased marks = 30.
The increase in marks \( = 30 - 20 = 10 \).
Percentage increase of marks \( = \frac { \text{Increase in Marks} }{ \text{Initial Marks} } \times 100\% \).
\( = \frac { 10 }{ 20 } \times 100\% \)
\( = \frac { 1 }{ 2 } \times 100\% = 50\% \).
In simple words: For the shirt, the price went down by Rs 20 from Rs 80, which is a 25% decrease. For the marks, they went up by 10 from 20, which is a 50% increase. To find the percentage change, we divide the change amount by the original amount and multiply by 100.

Exam Tip: The base for calculating percentage increase or decrease is always the *original* (initial) value. Use the formula: \( \frac{\text{Change}}{\text{Original Value}} \times 100\% \).

Question 2. My mother says, in her childhood petrol was Rs 1 a litre. It is Rs 52 per litre today. By what percentage has the price gone up?
Answer: The initial price of petrol was Rs 1 per litre.
The current price of petrol is Rs 52 per litre.
The increase in price \( = \text{Current Price} - \text{Initial Price} = \text{Rs } 52 - \text{Rs } 1 = \text{Rs } 51 \).
Percentage increase \( = \frac { \text{Increase in Price} }{ \text{Initial Price} } \times 100\% \).
\( = \frac { 51 }{ 1 } \times 100\% = 5100\% \).
Thus, the petrol price has increased by 5100%.
In simple words: The petrol price went from Rs 1 to Rs 52, which is a jump of Rs 51. To find the percentage increase, we compare this Rs 51 change to the original Rs 1, resulting in a 5100% increase.

Exam Tip: Always state the initial and final values clearly. For percentage increase, the base (denominator) for calculation is the initial value. Large percentage increases are possible when the initial value is very small.

Try These (Page 169)

Question 1. A shopkeeper bought a chair for Rs 375 and sold it for Rs 400. Find the gain percentage.
Answer: The Cost Price (CP) of the chair = Rs 375.
The Selling Price (SP) of the chair = Rs 400.
Since the Selling Price is greater than the Cost Price (SP > CP), there is a profit (gain).
Profit \( = \text{SP} - \text{CP} = \text{Rs } 400 - \text{Rs } 375 = \text{Rs } 25 \).
To find the gain percentage, we use the formula:
Profit % \( = \frac { \text{Profit} }{ \text{CP} } \times 100\% \).
\( = \frac { 25 }{ 375 } \times 100\% \).
\( = \frac { 1 }{ 15 } \times 100\% \).
\( = \frac { 20 }{ 3 }\% \).
\( = 6\frac { 2 }{ 3 }\% \).
Thus, the shopkeeper's gain percentage is \( 6\frac { 2 }{ 3 }\% \).
In simple words: The shopkeeper bought a chair for Rs 375 and sold it for Rs 400, making a profit of Rs 25. To get the percentage profit, we divide this profit by the original cost and multiply by 100, which gives about 6.67%.

Exam Tip: Remember that profit or loss percentage is always calculated based on the Cost Price (CP), unless specified otherwise. Make sure to simplify fractions before final percentage calculation.

Question 2. Cost of an item is Rs 50. It was sold with a profit of 12%. Find the selling price.
Answer: The Cost Price (CP) of the item = Rs 50.
The profit percentage is given as 12%.
First, calculate the actual profit amount:
Profit \( = 12\% \text{ of Rs } 50 \).
\( = \frac { 12 }{ 100 } \times 50 \).
\( = \frac { 12 }{ 2 } = \text{Rs } 6 \).
Now, calculate the Selling Price (SP) using the formula:
SP \( = \text{CP} + \text{Profit} \).
SP \( = \text{Rs } 50 + \text{Rs } 6 = \text{Rs } 56 \).
Thus, the selling price of the item is Rs 56.
In simple words: If an item costs Rs 50 and is sold for a 12% profit, first find 12% of Rs 50 (which is Rs 6). Then, add this profit to the original cost to get the selling price of Rs 56.

Exam Tip: To find the selling price with a given profit percentage, calculate the profit amount first (percentage of CP) and then add it to the CP. Be careful with calculations involving percentages.

Question 3. An article was sold for Rs 250 with a profit of 5%. What was its cost price?
Answer: Let the Cost Price (CP) of the article be \( x \).
The profit percentage is 5%.
The profit amount \( = 5\% \text{ of } x = \frac { 5 }{ 100 } \times x = \frac { x }{ 20 } \).
The Selling Price (SP) is the Cost Price plus the Profit:
SP \( = \text{CP} + \text{Profit} \).
We are given SP = Rs 250.
So, \( 250 = x + \frac { x }{ 20 } \).
To add \( x \) and \( \frac{x}{20} \), find a common denominator:
\( 250 = \frac { 20x }{ 20 } + \frac { x }{ 20 } \).
\( 250 = \frac { 21x }{ 20 } \).
Now, solve for \( x \):
\( x = 250 \times \frac { 20 }{ 21 } \).
\( x = \frac { 5000 }{ 21 } \).
\( x = 238\frac { 2 }{ 21 } \).
Thus, the cost price of the article is Rs \( 238\frac { 2 }{ 21 } \).
In simple words: We know the selling price (Rs 250) and the profit percentage (5%). If the cost price is \( x \), then the selling price is \( x \) plus 5% of \( x \). We set up an equation \( 250 = x + 0.05x \) or \( 250 = 1.05x \) and solve for \( x \).

Exam Tip: When given selling price and profit percentage, let the cost price be \( x \). Formulate an equation \( SP = CP + \text{Profit} \) (where profit is a percentage of CP) and solve for \( x \). This method is robust for such problems.

Question 4. An item was sold for Rs 540 at a loss of 5%. What was its cost price?
Answer: Let the Cost Price (CP) of the item be \( x \).
The loss percentage is 5%.
The loss amount \( = 5\% \text{ of } x = \frac { 5 }{ 100 } \times x = \frac { x }{ 20 } \).
The Selling Price (SP) is the Cost Price minus the Loss:
SP \( = \text{CP} - \text{Loss} \).
We are given SP = Rs 540.
So, \( 540 = x - \frac { x }{ 20 } \).
To subtract \( \frac{x}{20} \) from \( x \), find a common denominator:
\( 540 = \frac { 20x }{ 20 } - \frac { x }{ 20 } \).
\( 540 = \frac { 19x }{ 20 } \).
Now, solve for \( x \):
\( x = 540 \times \frac { 20 }{ 19 } \).
\( x = \frac { 10800 }{ 19 } \).
\( x = 568\frac { 8 }{ 19 } \).
Thus, the cost price of the item is Rs \( 568\frac { 8 }{ 19 } \).
In simple words: The item sold for Rs 540 after a 5% loss. If the original cost was \( x \), then the selling price is \( x \) minus 5% of \( x \). We set up the equation \( 540 = x - 0.05x \) or \( 540 = 0.95x \) and then solve for \( x \) to find the cost price.

Exam Tip: When given selling price and loss percentage, let the cost price be \( x \). Formulate an equation \( SP = CP - \text{Loss} \) (where loss is a percentage of CP) and solve for \( x \).

Try These (Page 170)

Question 1. Rs 10,000 is invested at 5% interest rate p.a. Find the interest at the end of one year.
Answer: We are given the following information:
Principal (P) = Rs 10,000.
Rate (R) = 5% p.a.
Time (T) = 1 year.
To find the simple interest, we use the formula:
Simple Interest \( = \frac { P \times R \times T }{ 100 } \).
\( = \frac { 10,000 \times 5 \times 1 }{ 100 } \).
\( = 100 \times 5 \times 1 \).
\( = \text{Rs } 500 \).
Thus, the interest at the end of one year will be Rs 500.
In simple words: To find the interest for one year, we multiply the principal amount (Rs 10,000) by the interest rate (5%) and the time (1 year), then divide by 100. This calculation gives Rs 500.

Exam Tip: Always remember the simple interest formula: \( SI = \frac{PRT}{100} \). Ensure that the rate (R) is in percentage per annum and time (T) is in years.

Question 2. Rs 3,500 is given at 7% p.a. rate of interest. Find the interest which will be received at the end of two years.
Answer: We have the following details:
Principal (P) = Rs 3,500.
Rate (R) = 7% p.a.
Time (T) = 2 years.
To calculate the simple interest, we use the formula:
Simple Interest \( = \frac { P \times R \times T }{ 100 } \).
\( = \frac { 3,500 \times 7 \times 2 }{ 100 } \).
\( = 35 \times 7 \times 2 \).
\( = 245 \times 2 \).
\( = \text{Rs } 490 \).
Thus, the interest received at the end of two years will be Rs 490.
In simple words: For a principal of Rs 3,500 at 7% interest over 2 years, we multiply 3500 by 7 and 2, then divide by 100. This calculation yields an interest of Rs 490.

Exam Tip: Ensure that all given values (Principal, Rate, Time) are substituted correctly into the simple interest formula. Simplify the numbers before multiplying for easier calculation.

Question 3. Rs 6,050 is borrowed at 6.5% rate of interest p.a. Find the interest and the amount to be paid at the end of 3 years.
Answer: We are given the following:
Principal (P) = Rs 6,050.
Rate (R) = 6.5% p.a.
Time (T) = 3 years.
To calculate the simple interest:
Simple Interest \( = \frac { P \times R \times T }{ 100 } \).
\( = \frac { 6,050 \times 6.5 \times 3 }{ 100 } \).
\( = \frac { 6,050 \times 65 \times 3 }{ 1,000 } \) (multiplying numerator and denominator by 10 to remove decimal).
\( = \frac { 1,179,750 }{ 1,000 } \).
\( = \text{Rs } 1179.75 \).
Now, to find the total amount to be paid:
Amount \( = \text{Principal} + \text{Interest} \).
\( = \text{Rs } 6,050 + \text{Rs } 1,179.75 \).
\( = \text{Rs } 7,229.75 \).
Thus, the total amount to be paid at the end of 3 years is Rs 7,229.75.
In simple words: For a loan of Rs 6,050 at 6.5% interest for 3 years, first find the simple interest, which is Rs 1,179.75. Then add this interest to the original loan amount to get the total amount to be repaid, which is Rs 7,229.75.

Exam Tip: When the rate has a decimal, it's often helpful to multiply the numerator and denominator by 10 (or 100, etc.) to convert the decimal to a whole number, simplifying calculations. Remember to add the calculated interest to the principal to find the total amount.

Question 4. Rs 7,000 is borrowed at 3.5% rate of interest p.a. borrowed for 2 years. Find the amount to be paid at the end of the second year.
Answer: We have the following given values:
Principal (P) = Rs 7,000.
Rate (R) = 3.5% p.a.
Time (T) = 2 years.
To calculate the simple interest:
Simple Interest \( = \frac { P \times R \times T }{ 100 } \).
\( = \frac { 7,000 \times 3.5 \times 2 }{ 100 } \).
\( = 70 \times 3.5 \times 2 \).
\( = 70 \times 7 \).
\( = \text{Rs } 490 \).
Now, to find the total amount to be paid:
Amount \( = \text{Principal} + \text{Interest} \).
\( = \text{Rs } 7,000 + \text{Rs } 490 \).
\( = \text{Rs } 7,490 \).
Thus, the amount to be paid at the end of the 2nd year is Rs 7,490.
In simple words: For a loan of Rs 7,000 at 3.5% interest for 2 years, first we calculate the interest, which is Rs 490. Then we add this interest to the initial loan amount to get the final amount to be paid, which is Rs 7,490.

Exam Tip: Always make sure to calculate both the simple interest and the total amount if the question asks for both. Be mindful of decimal points in the interest rate and handle them carefully during multiplication.

Try These (Page 171)

Question 1. You have Rs 2,400 in your account and the interest rate is 5%. After how many years would you earn 240 as interest.
Answer: We are given the following:
Principal (P) = Rs 2,400.
Rate (R) = 5% p.a.
Simple Interest (SI) = Rs 240.
Let the time be T years. We use the simple interest formula to find T:
Simple Interest \( = \frac { P \times R \times T }{ 100 } \).
Substituting the known values:
\( 240 = \frac { 2,400 \times 5 \times T }{ 100 } \).
\( 240 = 24 \times 5 \times T \).
\( 240 = 120 \times T \).
To solve for T:
\( T = \frac { 240 }{ 120 } \).
\( T = 2 \text{ years} \).
Thus, an interest of Rs 240 will be obtained after 2 years.
In simple words: If you have Rs 2,400 at 5% interest and want to earn Rs 240, you can use the simple interest formula to find the time. By plugging in the numbers, we discover it will take 2 years.

Exam Tip: When finding time (T) using the simple interest formula, rearrange the formula to \( T = \frac{SI \times 100}{P \times R} \). Ensure units are consistent (years for time, percentage for rate).

Question 2. On a certain sum the interest paid after 3 years is Rs 450 at 5% rate of interest per annum. Find the sum.
Answer: We are given the following details:
Simple Interest (SI) = Rs 450.
Time (T) = 3 years.
Rate (R) = 5% p.a.
Let the principal sum be P. We use the simple interest formula to find P:
Simple Interest \( = \frac { P \times R \times T }{ 100 } \).
Substituting the known values:
\( 450 = \frac { P \times 5 \times 3 }{ 100 } \).
\( 450 = \frac { 15P }{ 100 } \).
To solve for P:
\( P = 450 \times \frac { 100 }{ 15 } \).
\( P = 30 \times 100 \).
\( P = \text{Rs } 3000 \).
Thus, the required sum (Principal) is Rs 3000.
In simple words: If the interest earned is Rs 450 over 3 years at a 5% rate, we can use the simple interest formula to find the starting amount. By rearranging the formula and solving, we find the initial sum was Rs 3000.

Exam Tip: When finding the principal (P) using the simple interest formula, rearrange it to \( P = \frac{SI \times 100}{R \times T} \). This helps in directly calculating the sum without extra steps.