Class 7 Maths - Comparing Quantities NCERT Solutions

Chapter 8: Comparing Quantities (NCERT Solutions)

Exercise 8.1

Q1. Find the ratio of:
(a) ₹ 5 to 50 paise
(b) 15 kg to 210 g
(c) 9 m to 27 cm
(d) 30 days to 36 hours

To find the ratio, both quantities must be in the same unit.

(a) ₹ 5 = 5 × 100 paise = 500 paise.
Ratio = 500 / 50 = 10 / 1 = 10:1

(b) 15 kg = 15 × 1000 g = 15000 g.
Ratio = 15000 / 210 = 1500 / 21 = (1500 ÷ 3) / (21 ÷ 3) = 500 / 7 = 500:7

(c) 9 m = 9 × 100 cm = 900 cm.
Ratio = 900 / 27 = (900 ÷ 9) / (27 ÷ 9) = 100 / 3 = 100:3

(d) 30 days = 30 × 24 hours = 720 hours.
Ratio = 720 / 36 = (720 ÷ 36) / (36 ÷ 36) = 20 / 1 = 20:1

Q2. In a computer lab, there are 3 computers for every 6 students. How many computers will be needed for 24 students?

Using the unitary method:

Number of computers for 6 students = 3
Number of computers for 1 student = 3 / 6 = ½
Number of computers for 24 students = ½ × 24 = 12 computers

Q3. Population of Rajasthan = 570 lakhs and population of UP = 1660 lakhs.
Area of Rajasthan = 3 lakh km² and area of UP = 2 lakh km².
(i) How many people are there per km² in both these States?
(ii) Which State is less populated?

(i) People per km² = Population / Area
For Rajasthan:
People per km² = 570 lakhs / 3 lakh km² = 190 people per km²
For UP:
People per km² = 1660 lakhs / 2 lakh km² = 830 people per km²

(ii) Comparing the two states, Rajasthan has 190 people per km² while UP has 830 people per km².
Therefore, Rajasthan is less populated.

Exercise 8.2

Q1. Convert the given fractional numbers to per cents.
(a) 1/8 (b) 5/4 (c) 3/40 (d) 2/7

To convert a fraction to a percentage, multiply by 100%.

(a) (1/8) × 100% = 100/8 % = 25/2 % = 12.5% or 12½%

(b) (5/4) × 100% = 5 × 25% = 125%

(c) (3/40) × 100% = 300/40 % = 30/4 % = 15/2 % = 7.5%

(d) (2/7) × 100% = 200/7 % = 28&frac4;7% (or approx 28.57%)

Q2. Convert the given decimal fractions to per cents.
(a) 0.65 (b) 2.1 (c) 0.02 (d) 12.35

(a) 0.65 = (65/100) × 100% = 65%

(b) 2.1 = (21/10) × 100% = 21 × 10% = 210%

(c) 0.02 = (2/100) × 100% = 2%

(d) 12.35 = (1235/100) × 100% = 1235%

Q3. Estimate what part of the figures is coloured and hence find the per cent which is coloured.

(Assuming standard textbook figures):

(i) The figure is divided into 4 equal parts, and 1 part is coloured.
Fraction = 1/4
Percentage = (1/4) × 100% = 25%

(ii) The figure is divided into 5 equal parts, and 3 parts are coloured.
Fraction = 3/5
Percentage = (3/5) × 100% = 3 × 20% = 60%

(iii) The figure is divided into 8 equal parts, and 3 parts are coloured.
Fraction = 3/8
Percentage = (3/8) × 100% = 300/8 % = 75/2 % = 37.5%

Q4. Find:
(a) 15% of 250 (b) 1% of 1 hour (c) 20% of ₹ 2500 (d) 75% of 1 kg

(a) (15/100) × 250 = 15 × 2.5 = 37.5

(b) 1 hour = 60 minutes.
1% of 60 mins = (1/100) × 60 = 0.6 minutes.
Or in seconds: (1/100) × 3600 seconds = 36 seconds.

(c) (20/100) × 2500 = 20 × 25 = ₹ 500

(d) 1 kg = 1000 g.
(75/100) × 1000 g = 75 × 10 = 750 g (or 0.75 kg).

Q5. Find the whole quantity if:
(a) 5% of it is 600
(b) 12% of it is ₹ 1080
(c) 40% of it is 500 km
(d) 70% of it is 14 minutes
(e) 8% of it is 40 litres

Let the whole quantity be x.

(a) 5% of x = 600 ⇒ (5/100) × x = 600 ⇒ x = (600 × 100) / 5 = 120 × 100 = 12000

(b) 12% of x = 1080 ⇒ (12/100) × x = 1080 ⇒ x = (1080 × 100) / 12 = 90 × 100 = ₹ 9000

(c) 40% of x = 500 ⇒ (40/100) × x = 500 ⇒ x = (500 × 100) / 40 = 5000 / 4 = 1250 km

(d) 70% of x = 14 ⇒ (70/100) × x = 14 ⇒ x = (14 × 100) / 70 = 1400 / 70 = 20 minutes

(e) 8% of x = 40 ⇒ (8/100) × x = 40 ⇒ x = (40 × 100) / 8 = 5 × 100 = 500 litres

Q6. Convert given per cents to decimal fractions and also to fractions in simplest forms:
(a) 25% (b) 150% (c) 20% (d) 5%

Percentage	Fraction (Simplest Form)	Decimal
(a) 25%	25/100 = 1/4	25/100 = 0.25
(b) 150%	150/100 = 15/10 = 3/2	150/100 = 1.5
(c) 20%	20/100 = 1/5	20/100 = 0.2
(d) 5%	5/100 = 1/20	5/100 = 0.05

Q7. In a city, 30% are females, 40% are males and remaining are children. What per cent are children?

Total percentage of city population = 100%
Percentage of females = 30%
Percentage of males = 40%
Percentage of children = Total - (Females + Males)
Percentage of children = 100% - (30% + 40%) = 100% - 70% = 30%

Q8. Out of 15,000 voters in a constituency, 60% voted. Find the percentage of voters who did not vote. Can you now find how many actually did not vote?

Total voters = 15,000
Percentage of voters who voted = 60%
Percentage of voters who did not vote = 100% - 60% = 40%

Number of people who did not vote = 40% of 15,000
= (40/100) × 15,000 = 40 × 150 = 6,000

Q9. Meeta saves ₹ 4000 from her salary. If this is 10% of her salary. What is her salary?

Let Meeta's total salary be x.
10% of x = ₹ 4000
(10/100) × x = 4000
x/10 = 4000
x = 4000 × 10 = ₹ 40000

Q10. A local cricket team played 20 matches in one season. It won 25% of them. How many matches did they win?

Total matches played = 20
Matches won = 25% of 20
= (25/100) × 20
= 1/4 × 20 = 5 matches

Exercise 8.3

Q1. Tell what is the profit or loss in the following transactions. Also find profit per cent or loss per cent in each case.
(a) Gardening shears bought for ₹ 250 and sold for ₹ 325.
(b) A refrigerator bought for ₹ 12,000 and sold at ₹ 13,500.
(c) A cupboard bought for ₹ 2,500 and sold at ₹ 3,000.
(d) A skirt bought for ₹ 250 and sold at ₹ 150.

(a) CP = ₹ 250, SP = ₹ 325
Since SP > CP, it's a Profit.
Profit = SP - CP = 325 - 250 = ₹ 75
Profit % = (Profit/CP) × 100 = (75/250) × 100 = 30%

(b) CP = ₹ 12,000, SP = ₹ 13,500
Profit = SP - CP = 13500 - 12000 = ₹ 1500
Profit % = (1500/12000) × 100 = (1/8) × 100 = 12.5%

(c) CP = ₹ 2,500, SP = ₹ 3,000
Profit = SP - CP = 3000 - 2500 = ₹ 500
Profit % = (500/2500) × 100 = (1/5) × 100 = 20%

(d) CP = ₹ 250, SP = ₹ 150
Since CP > SP, it's a Loss.
Loss = CP - SP = 250 - 150 = ₹ 100
Loss % = (Loss/CP) × 100 = (100/250) × 100 = 40%

Q2. Convert each part of the ratio to percentage:
(a) 3:1 (b) 2:3:5 (c) 1:4 (d) 1:2:5

(a) Ratio = 3:1
Total parts = 3 + 1 = 4
Percentage of 1st part = (3/4) × 100 = 75%
Percentage of 2nd part = (1/4) × 100 = 25%

(b) Ratio = 2:3:5
Total parts = 2 + 3 + 5 = 10
1st part = (2/10) × 100 = 20%
2nd part = (3/10) × 100 = 30%
3rd part = (5/10) × 100 = 50%

(c) Ratio = 1:4
Total parts = 1 + 4 = 5
1st part = (1/5) × 100 = 20%
2nd part = (4/5) × 100 = 80%

(d) Ratio = 1:2:5
Total parts = 1 + 2 + 5 = 8
1st part = (1/8) × 100 = 12.5%
2nd part = (2/8) × 100 = 25%
3rd part = (5/8) × 100 = 62.5%

Q3. The population of a city decreased from 25,000 to 24,500. Find the percentage decrease.

Initial Population = 25,000
Final Population = 24,500
Decrease in Population = 25000 - 24500 = 500
Percentage Decrease = (Decrease / Initial) × 100
= (500 / 25000) × 100 = 2%

Q4. Arun bought a car for ₹ 3,50,000. The next year, the price went upto ₹ 3,70,000. What was the Percentage of price increase?

Initial Price = ₹ 3,50,000
Final Price = ₹ 3,70,000
Increase in Price = 3,70,000 - 3,50,000 = ₹ 20,000
Percentage Increase = (Increase / Initial) × 100
= (20000 / 350000) × 100 = (2/35) × 100 = 200/35 = 5&frac5;7%

Q5. I buy a T.V. for ₹ 10,000 and sell it at a profit of 20%. How much money do I get for it?

Cost Price (CP) = ₹ 10,000
Profit % = 20%
Profit = 20% of 10,000 = (20/100) × 10000 = ₹ 2000
Selling Price (SP) = CP + Profit = 10000 + 2000 = ₹ 12,000

Q6. Juhi sells a washing machine for ₹ 13,500. She loses 20% in the bargain. What was the price at which she bought it?

Selling Price (SP) = ₹ 13,500
Loss % = 20%
Let the Cost Price be CP.
Loss = 20% of CP = 0.20 × CP
SP = CP - Loss
13500 = CP - 0.20 CP
13500 = 0.80 CP
CP = 13500 / 0.80 = ₹ 16,875

Q7. (i) Chalk contains calcium, carbon and oxygen in the ratio 10:3:12. Find the percentage of carbon in chalk.
(ii) If in a stick of chalk, carbon is 3g, what is the weight of the chalk stick?

(i) Ratio = 10:3:12
Total parts = 10 + 3 + 12 = 25
Part of Carbon = 3 / 25
Percentage of Carbon = (3/25) × 100 = 3 × 4 = 12%

(ii) Let total weight be x.
We know carbon is 12% of the total weight.
12% of x = 3g
(12/100) × x = 3
x = (3 × 100) / 12 = 300 / 12 = 25 g

Q8. Amina buys a book for ₹ 275 and sells it at a loss of 15%. How much does she sell it for?

CP = ₹ 275
Loss % = 15%
Loss = 15% of 275 = (15/100) × 275 = ₹ 41.25
Selling Price (SP) = CP - Loss = 275 - 41.25 = ₹ 233.75

Q9. Find the amount to be paid at the end of 3 years in each case:
(a) Principal = ₹ 1,200 at 12% p.a.
(b) Principal = ₹ 7,500 at 5% p.a.

Time (T) = 3 years. Simple Interest (SI) = (P × R × T) / 100

(a) P = ₹ 1,200, R = 12%
SI = (1200 × 12 × 3) / 100 = 12 × 36 = ₹ 432
Amount = P + SI = 1200 + 432 = ₹ 1,632

(b) P = ₹ 7,500, R = 5%
SI = (7500 × 5 × 3) / 100 = 75 × 15 = ₹ 1125
Amount = P + SI = 7500 + 1125 = ₹ 8,625

Q10. What rate gives ₹ 280 as interest on a sum of ₹ 56,000 in 2 years?

SI = ₹ 280
P = ₹ 56,000
T = 2 years
R = (100 × SI) / (P × T)
R = (100 × 280) / (56000 × 2) = 28000 / 112000 = 1/4 = 0.25%

Q11. If Meena gives an interest of ₹ 45 for one year at 9% rate p.a.. What is the sum she has borrowed?

SI = ₹ 45
R = 9%
T = 1 year
P = (100 × SI) / (R × T)
P = (100 × 45) / (9 × 1) = 4500 / 9 = ₹ 500

Class 7 Maths - Comparing Quantities Practice

Chapter 8: Comparing Quantities (Practice Questions)

RD Sharma / Extra Practice

Q1. The ratio of two numbers is 3:4. If the sum of the numbers is 63, find both the numbers.

Let the numbers be 3x and 4x.
Given, 3x + 4x = 63 ⇒ 7x = 63 ⇒ x = 9
First number = 3 × 9 = 27
Second number = 4 × 9 = 36

Q2. Convert 7:8 into a percentage.

7:8 = 7/8
Percentage = (7/8) × 100% = 700/8 % = 87.5%

Q3. A mixture contains milk and water in the ratio 3:2. Find the percentage of milk in the mixture.

Total parts = 3 + 2 = 5
Parts of milk = 3
Percentage of milk = (3/5) × 100 = 3 × 20 = 60%

Q4. Express 0.045 as a percentage.

0.045 = (45 / 1000) × 100% = 45 / 10 % = 4.5%

Q5. What is 120% of ₹ 450?

(120 / 100) × 450 = 12 × 45 = ₹ 540

Q6. If 15% of a number is 45, what is the number?

Let the number be x.
15% of x = 45
(15 / 100) × x = 45
x = (45 × 100) / 15 = 3 × 100 = 300

Q7. The price of an article increased from ₹ 40 to ₹ 50. What is the percentage increase?

Increase = 50 - 40 = 10
% Increase = (Increase / Original Price) × 100
= (10 / 40) × 100 = (1 / 4) × 100 = 25%

Q8. Out of a salary of ₹ 25,000, a man spends 30% on food, 20% on rent, and saves the rest. What is his saving?

Total spend percentage = 30% + 20% = 50%
Savings percentage = 100% - 50% = 50%
Savings = 50% of 25,000 = (50/100) × 25000 = 1/2 × 25000 = ₹ 12,500

Q9. By selling a chair for ₹ 450, a shopkeeper loses 10%. Find the cost price of the chair.

SP = ₹ 450
Loss = 10%
Let CP be x.
SP = CP - Loss
450 = x - (10/100)x
450 = x - 0.1x = 0.9x
x = 450 / 0.9 = ₹ 500

Q10. An article is bought for ₹ 300 and sold at a profit of 15%. Find the selling price.

CP = ₹ 300
Profit = 15% of 300 = (15/100) × 300 = ₹ 45
SP = CP + Profit = 300 + 45 = ₹ 345

Q11. A man earns a profit of 25% by selling a fan for ₹ 1500. Find the cost price.

Let CP be x.
SP = CP + Profit
1500 = x + 0.25x
1500 = 1.25x
x = 1500 / 1.25 = 150000 / 125 = ₹ 1200

Q12. What sum will become ₹ 540 in 3 years at 5% per annum simple interest?

Let Principal be P.
Rate = 5%, Time = 3 years, Amount = 540.
Amount = P + SI = P + (P × R × T)/100
540 = P + (P × 5 × 3) / 100
540 = P + 15P/100 = P + 0.15P = 1.15P
P = 540 / 1.15 = 54000 / 115 = ₹ 469.56 (approx)

Q13. In how many years will ₹ 800 become ₹ 920 at 5% p.a. simple interest?

P = 800, Amount = 920, R = 5%
SI = Amount - P = 920 - 800 = 120.
Time = (100 × SI) / (P × R)
Time = (100 × 120) / (800 × 5) = 12000 / 4000 = 3 years

Q14. Find the simple interest on ₹ 2500 for 2 years 6 months at 6% p.a.

P = 2500, R = 6%
Time = 2 years 6 months = 2 + 6/12 = 2.5 years (or 5/2 years).
SI = (P × R × T) / 100
SI = (2500 × 6 × 2.5) / 100 = 25 × 15 = ₹ 375

Q15. At what rate per cent per annum will a sum double itself in 8 years?

Let Principal be P.
Amount = 2P.
Interest (SI) = 2P - P = P.
Time = 8 years.
Rate = (100 × SI) / (P × T) = (100 × P) / (P × 8) = 100 / 8 = 12.5% = 12½%

Q16. If 20% of a = b, then b% of 20 is the same as:

20% of a = b ⇒ b = 0.20a
We need b% of 20 = (b/100) × 20 = b/5
Substitute b = 0.20a:
b/5 = 0.20a / 5 = 0.04a = 4% of a.

Q17. The ratio of boys to girls in a class is 5:3. If the total number of students is 48, find the number of girls.

Total parts = 5 + 3 = 8.
Total students = 48.
Value of 1 part = 48 / 8 = 6.
Number of girls = 3 parts = 3 × 6 = 18 girls.

Q18. If the cost of 5 kg of apples is ₹ 450, what is the cost of 12 kg of apples?

Cost of 5 kg = ₹ 450
Cost of 1 kg = 450 / 5 = ₹ 90
Cost of 12 kg = 90 × 12 = ₹ 1080

Q19. Find the simple interest if Principal is ₹ 400, Rate is 3 paise per rupee per annum for 4 years.

Rate = 3 paise per rupee = 3 / 100 = 3% p.a.
SI = (P × R × T) / 100 = (400 × 3 × 4) / 100 = 4 × 12 = ₹ 48

Q20. A person sells an article for ₹ 550, gaining 1/10 of its C.P. Find his gain %.

Gain = 1/10 of CP = 0.1 CP
SP = CP + Gain = CP + 0.1 CP = 1.1 CP
We know SP = 550.
1.1 CP = 550 ⇒ CP = 550 / 1.1 = 500.
Gain = 550 - 500 = 50.
Gain % = (Gain / CP) × 100 = (50 / 500) × 100 = 10%

(Or directly, since gain is 1/10 of CP, gain% is 1/10 × 100 = 10%)

Class 7 Maths - Comparing Quantities Summary

Chapter 8: Comparing Quantities (Concepts & Summary)

1. Ratio & Proportion

Ratio: A ratio compares two quantities of the same kind. To compare two quantities, their units must be the same.
E.g., Ratio of 5 km to 500 m = (5000 m / 500 m) = 10:1.
Equivalent Ratios: Two ratios can be compared by converting them to like fractions. If these fractions are equal, the given ratios are equivalent.
Proportion: When two ratios are equal, they are said to be in proportion.
If a/b = c/d, then a, b, c, d are in proportion (written as a:b :: c:d).
Unitary Method: The method in which the value of a single unit is found first, and then the value of the required number of units.

2. Percentage

Meaning: Percentage means 'per hundred'. It is denoted by the symbol '%'. 1% means 1 out of 100 or 1/100.
Fractions to %: Multiply the fraction by 100 and add the % sign.
E.g., 3/4 = (3/4 × 100)% = 75%
Decimals to %: Shift the decimal point two places to the right.
E.g., 0.45 = 45%
% to Fraction/Decimal: Divide by 100.
E.g., 20% = 20/100 = 1/5 = 0.2

3. Profit and Loss

Cost Price (CP): Price at which an article is bought.
Selling Price (SP): Price at which an article is sold.
Profit: If SP > CP, then Profit = SP - CP.
Loss: If CP > SP, then Loss = CP - SP.
Profit %: = (Profit / CP) × 100
Loss %: = (Loss / CP) × 100
Note: Profit and Loss percentages are always calculated on the Cost Price (CP).

4. Simple Interest

Principal (P): The money borrowed or lent out for a certain period.
Rate of Interest (R): The interest charged on ₹ 100 for one year (expressed as % p.a.).
Time (T): The duration for which the money is borrowed.
Simple Interest (SI): The extra money paid for using others' money.
Formula: SI = (P × R × T) / 100
Amount (A): The total money paid back.
Formula: Amount = Principal + Simple Interest (A = P + SI)

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