Comparing Quantities is a fundamental concept in mathematics that deals with understanding and analyzing different quantities. This chapter introduces you to various methods and tools to compare numbers, values, and proportions.
Key Topics:
- Ratio: A comparison of two quantities of the same kind.
- Proportion: A statement of equality between two ratios.
- Percentage: A way of expressing a number as a fraction of 100.
- Profit and Loss: Calculating profit or loss in business transactions.
- Simple Interest: Understanding the concept of interest and calculating simple interest.
Important Concepts:
- Equivalent Ratios: Different ratios that represent the same comparison.
- Unitary Method: Finding the value of one unit and then multiplying it to find the value of the required number of units.
- Discount and Markup: Understanding price reductions and increases.
- Sales Tax: Calculating the amount of tax added to the cost price.
- Compound Interest: Understanding how interest is calculated on both the principal and accumulated interest.
By mastering these concepts, you can effectively compare and analyze different quantities in various real-life situations.
Exercise 7.1
1. Convert the given fractional numbers to per cents:
Ans :
(a) 1/8 as a percentage:
(1/8)*100 = 12.5%
(b) 5/4 as a percentage:
(5/4)*100 = 125%
(c) 3/40 as a percentage:
(3/40)*100 = 7.5%
(d) 2/7 as a percentage:
(2/7)*100 = 28.57%
2. Convert the given decimal fractions to per cents:
(а) 0.65
(b) 2.1
(c) 0.02
(d) 12.35
Ans :
(a) 0.65
- 0.65 * 100 = 65%
(b) 2.1
- 2.1 * 100 = 210%
(c) 0.02
- 0.02 * 100 = 2%
(d) 12.35
- 12.35 * 100 = 1235%
3. Estimate what part of the figures is coloured and hence find the per cent which is coloured.
Ans :
Figure (i)
- The figure is divided into 4 equal parts.
- 1 out of 4 parts is colored.
Calculation:
- Fraction of colored part = 1/4
- Percentage of colored part = (1/4) * 100% = 25%
Figure (ii)
- The figure is divided into 5 equal parts.
- 3 out of 5 parts are colored.
Calculation:
- Fraction of colored part = 3/5
- Percentage of colored part = (3/5) * 100% = 60%
Figure (iii)
- The figure is divided into 8 equal parts.
- 3 out of 8 parts are colored.
Calculation:
- Fraction of colored part = 3/8
- Percentage of colored part = (3/8) * 100% = 37.5%
4. Find:
(a) 15% of 250
(b) 1% of 1 hour
(c) 20% of ₹ 2500
(d) 75% of 1 kg
Ans :
(a) 15% of 250
- 15% of 250 = (15/100) * 250 = 37.5
(b) 1% of 1 hour
- 1 hour = 60 minutes
- 1% of 60 minutes = (1/100) * 60 = 0.6 minutes = 36 seconds
(c) 20% of ₹2500
- 20% of ₹2500 = (20/100) * 2500 = ₹500
(d) 75% of 1 kg
- 1 kg = 1000 grams
- 75% of 1000 grams = (75/100) * 1000 = 750 grams
5. 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
Ans :
Let’s denote the whole quantity as ‘x’.
a) 5% of it is 600
- 5% of x = 600
- (5/100) * x = 600
- x = (600 * 100) / 5
- x = 12000
b) 12% of it is 1080
- 12% of x = 1080
- (12/100) * x = 1080
- x = (1080 * 100) / 12
- x = 9000
c) 40% of it is 500 km
- 40% of x = 500
- (40/100) * x = 500
- x = (500 * 100) / 40
- x = 1250 km
d) 70% of it is 14 minutes
- 70% of x = 14
- (70/100) * x = 14
- x = (14 * 100) / 70
- x = 20 minutes
e) 8% of it is 40 litres
- 8% of x = 40
- (8/100) * x = 40
- x = (40 * 100) / 8
- x = 500 litres
6. Convert given per cents to decimal fractions and also to fractions in simplest forms:
(a) 25%
(b) 150%
(c) 20%
(d) 5%
Ans :
a) 25%
- Decimal: 25/100 = 0.25
- Fraction: 25/100 = 1/4
b) 150%
- Decimal: 150/100 = 1.5
- Fraction: 150/100 = 3/2
c) 20%
- Decimal: 20/100 = 0.2
- Fraction: 20/100 = 1/5
d) 5%
- Decimal: 5/100 = 0.05
- Fraction: 5/100 = 1/20
7. In a city, 30% are females, 40% are males and remaining are children. What per cent are children?
Ans :
Total percentage of people in a city = 100%
Percentage of females = 30%
Percentage of males = 40%
So, the percentage of children = Total percentage – (Percentage of females + Percentage of males)
= 100% – (30% + 40%) = 100% – 70%
= 30%
8. 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?
Ans :
Total number of voters = 15,000
Percentage of voters who voted = 60%
Therefore, the percentage of voters who did not vote = (100% – 60%) = 40%
Actual number of voters who did not vote = 40% of 15,000
= 40/100×15,000
= 6,000
9. Meena saves ₹ 400 from her salary. If this is 10% of her salary. What is her salary?
Ans :
- Meena saves ₹400.
- This saving is 10% of her total salary.
- We need to find her total salary.
Solution: If ₹400 is 10% of her salary, then:
- 1% of her salary = ₹400 / 10 = ₹40
- 100% of her salary (which is her total salary) = ₹40 * 100 = ₹4000
So, Meena’s salary is ₹4000.
10. A local cricket team played 20 matches in one season. It won 25% of them. How many matches did they win?
Ans :
Total matches played = 20Win percentage = 25%
To find the number of matches won, we need to find 25% of 20.
- 25% of 20 = (25/100) * 20 = 5
So, the team won 5 matches.
Exercise 7.2
1. 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.
Ans :
| Item | Profit/Loss | Profit/Loss Percentage |
| Gardening shears | Profit ₹75.00 | 30.00% |
| Refrigerator | Profit ₹1500.00 | 12.50% |
| Cupboard | Profit ₹500.00 | 20.00% |
| Skirt | Loss ₹100.00 | 40.00% |
2. Convert each part of the ratio to Percentage:
(a) 3:1
(b) 2:3:5
(c) 1 : 4
(d) 1:2:5
Ans :
(a) 3:1
- Total parts = 3 + 1 = 4
- First part as a percentage = (3/4) * 100 = 75%
- Second part as a percentage = (1/4) * 100 = 25%
(b) 2:3:5
- Total parts = 2 + 3 + 5 = 10
- First part as a percentage = (2/10) * 100 = 20%
- Second part as a percentage = (3/10) * 100 = 30%
- Third part as a percentage = (5/10) * 100 = 50%
(c) 1:4
- Total parts = 1 + 4 = 5
- First part as a percentage = (1/5) * 100 = 20%
- Second part as a percentage = (4/5) * 100 = 80%
(d) 1:2:5
- Total parts = 1 + 2 + 5 = 8
- First part as a percentage = (1/8) * 100 = 12.5%
- Second part as a percentage = (2/8) * 100 = 25%
- Third part as a percentage = (5/8) * 100 = 62.5%
3. The population of a city decreased from 25,000 to 24,500. Find the Percentage decrease.
Ans :
Step 1: Find the decrease in population
- Decrease = Original population – New population = 25000 – 24500 = 500
Step 2: Calculate the percentage decrease
- Percentage decrease = (Decrease / Original population) * 100 = (500 / 25000) * 100 = 0.02 * 100 = 2%
Therefore, the percentage decrease in population is 2%.
4. 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?
Ans :
Understanding the problem:
- Original price of the car = ₹3,50,000
- Price after one year = ₹3,70,000
Solution:
- Find the increase in price:
- Increase in price = New price – Original price = ₹3,70,000 – ₹3,50,000 = ₹20,000
- Calculate the percentage increase:
- Percentage increase = (Increase in price / Original price) * 100 = (₹20,000 / ₹3,50,000) * 100 = (1/17.5) * 100 ≈ 5.71%
Therefore, the percentage increase in the price of the car is approximately 5.71%.
5. I buy a TV for ₹ 10,000 and sell it at a profit of 20%. How much money do I get for it?
Ans :
Cost price of the TV = ₹10,000
Profit percentage = 20%
Solution:
- Calculate the profit amount:
- Profit = (Profit percentage / 100) * Cost price = (20/100) * ₹10,000 = ₹2,000
- Calculate the selling price:
- Selling price = Cost price + Profit = ₹10,000 + ₹2,000 = ₹12,000
Therefore, you get ₹12,000 for the TV.
6. Juhi sells a washing machine for ₹ 13,500. She loses 20% in the bargain. What was the price at which she bought it?
Ans :
Selling price (SP) = ₹13,500
Loss percentage = 20%
We need to find the cost price (CP)
Solution:
- If she loses 20%, it means the selling price is 80% of the cost price.
- So, 80% of CP = ₹13,500
To find 100% (which is the CP), we can use the following:
- CP = (100 / 80) * ₹13,500 = (5/4) * ₹13,500 = ₹16,875
Therefore, Juhi bought the washing machine for ₹16,875.
7. (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 3 g, what is the weight of the chalk stick?
Ans :
(i) Percentage of carbon in chalk
Understanding the problem:
- Ratio of calcium, carbon, and oxygen in chalk is 10:3:12.
- We need to find the percentage of carbon in chalk.
Solution:
- Total parts in the ratio = 10 + 3 + 12 = 25
- Carbon parts = 3
- Percentage of carbon = (Carbon parts / Total parts) * 100 = (3/25) * 100 = 12%
Therefore, the percentage of carbon in chalk is 12%.
(ii) Weight of the chalk stick
Understanding the problem:
- Carbon in a chalk stick = 3g
- Percentage of carbon in chalk = 12% (from part (i))
- We need to find the total weight of the chalk stick.
Solution:
- If 12% of the chalk stick is 3g, then 1% is 3g / 12 = 0.25g.
- So, 100% of the chalk stick (total weight) = 0.25g * 100 = 25g.
Therefore, the weight of the chalk stick is 25g.
8. Amina buys a book for ₹ 275 and sells it at a loss of 15%. How much does she sell it for?
Ans :
Cost price of the book = ₹275
Loss percentage = 15%
Solution:
- Calculate the loss amount:
- Loss = (Loss percentage / 100) * Cost price = (15/100) * ₹275 = ₹41.25
- Calculate the selling price:
- Selling price = Cost price – Loss = ₹275 – ₹41.25 = ₹233.75
Therefore, Amina sells the book for ₹233.75.
9. Find the amount to be paid at the end of 3 years in each case.
(a) Principal = ₹ 1200 at 12% p.a.
(b) Principal = ₹ 7500 at 5% p.a.
Ans :
| Principal | Rate | Amount (after 3 years) |
| ₹ 1200 | 12% | ₹ 1632.00 |
| ₹ 7500 | 5% | ₹ 8625.00 |
10. What rate gives ₹ 280 as interest on a sum of ₹ 56,000 in 2 years?
Ans :
Principal (P) = ₹56,000
Simple Interest (SI) = ₹280
Time (T) = 2 years
We need to find the Rate of Interest (R)
Solution:
We know the formula for Simple Interest (SI):
- SI = (P * R * T) / 100
We can rearrange the formula to find R:
- R = (SI * 100) / (P * T)
Substituting the given values:
- R = (280 * 100) / (56000 * 2) = 28000 / 112000 = 0.25
Therefore, the rate of interest is 0.25%.
11. If Meena gives an interest of ₹ 45 for one year at 9% rate p.a. What is the sum she has borrowed?
Ans :
Interest (SI) = ₹45
Rate (R) = 9% per year
Time (T) = 1 year
We need to find the Principal (P)
Solution:
We know the formula for Simple Interest (SI):
- SI = (P * R * T) / 100
We can rearrange the formula to find P:
- P = (SI * 100) / (R * T)
Substituting the given values:
- P = (45 * 100) / (9 * 1) = 4500 / 9 = ₹500
Therefore, Meena borrowed a sum of ₹500
