Important Questions on Comparing Quantities for Class 8

Important Questions on Comparing Quantities for Class 8 are available in this Maths article. Comparing Quantities Class 8 Important Questions are very useful to solve the problems easily. This article helps the students to know the key questions and answers about Comparing Quantities. Comparing quantities involves ratios, percentages, profit/loss, discounts, sales tax, and compound interest to analyze proportional relationships. Our subject experts have provided detailed solutions for these problems based on the CBSE syllabus and the NCERT textbook. This material helps students revise the chapter easily and perform well in the final examination. 

Table of Contents

• Exercise 8.1: Ratios and Proportions
• Exercise 8.2: Understanding Percentages
• Exercise 8.3: Discount and Simple Interest
• Mixed Exercise Problems
• Tips for Understanding Comparing Quantities
• Most Common Examination Questions (Board Exams)

Exercise 8.1: Ratios and Proportions

Question 1: What is a Ratio? How Do We Write It?

Answer: A ratio is a comparison between two quantities of the same kind. It shows how many times one quantity is greater than or smaller than another quantity.

How to write ratios: A ratio between two quantities a and b is written as a:b (read as "a is to b")

We can also write it as a fraction: a/b

Examples of ratios:

• 2:3 means for every 2 of one thing, there are 3 of another

• 5:10 means 5 compared to 10

• 3:4 means for every 3 items of one kind, there are 4 of another kind

Question 2: Simplify the Ratio 20:30

Solution: To simplify a ratio, we divide both parts by their Greatest Common Divisor (GCD).

20:30

First, find the GCD of 20 and 30:

• Factors of 20: 1, 2, 4, 5, 10, 20

• Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30

• Common factors: 1, 2, 5, 10

• Greatest common factor: 10

Now divide both by 10: 20 ÷ 10 : 30 ÷ 10 = 2:3

Answer: 20:30 = 2:3 (simplified form)

This means the original ratio simplified is 2:3.

Question 3: What is a Proportion? When Are Two Ratios Equal?

Answer: A proportion is when two ratios are equal to each other. We say that four quantities are in proportion if the ratio between the first two is equal to the ratio between the last two.

How to write proportions: If a:b = c:d, we write: a:b::c:d (read as "a is to b as c is to d")

When are two ratios equal? Two ratios a:b and c:d are equal if: a/b = c/d

This also means: a × d = b × c (cross multiplication)

Examples:

• 2:3 = 4:6 because 2/3 = 4/6 (both equal 0.67)

• 1:2 = 5:10 because 1/2 = 5/10 = 0.5

• 3:4 = 6:8 because 3 × 8 = 4 × 6 (both equal 24)

Question 4: Are 4:6 and 2:3 in Proportion? Check Using Cross Multiplication

Solution: Two ratios a:b and c:d are in proportion if a × d = b × c

For 4:6 and 2:3: a = 4, b = 6, c = 2, d = 3

Cross multiplication: a × d = 4 × 3 = 12 b × c = 6 × 2 = 12

Since 12 = 12, the ratios are in proportion.

Answer: Yes, 4:6 and 2:3 are in proportion.

Question 5: If 10 Notebooks Cost Rs. 150, What is the Cost of 25 Notebooks?

Solution: This is a proportion problem. As the number of notebooks increases, the cost increases at the same rate.

Given: 10 notebooks cost Rs. 150

Method 1: Using ratio Cost per notebook = 150 ÷ 10 = Rs. 15 per notebook Cost of 25 notebooks = 25 × 15 = Rs. 375

Method 2: Using proportion If 10 notebooks cost Rs. 150, then 25 notebooks cost x Rs.

10:150 = 25:x

Using cross multiplication: 10 × x = 150 × 25 10x = 3750 x = 3750 ÷ 10 x = 375

Answer: 25 notebooks cost Rs. 375

Question 6: What is the Ratio of 45 Minutes to 1 Hour?

Solution: First, we need to convert both to the same unit.

1 hour = 60 minutes

Now find the ratio: 45 minutes : 60 minutes

Simplify by dividing by GCD: GCD of 45 and 60 = 15

45 ÷ 15 : 60 ÷ 15 = 3:4

Answer: The ratio of 45 minutes to 1 hour is 3:4

Exercise 8.2: Understanding Percentages

Question 7: What is a Percentage? How Do We Calculate It?

Answer: A percentage is a way of expressing a number as a fraction of 100. The word "percent" means "out of 100" or "per hundred."

A percentage is written with the symbol % (percent sign).

How to calculate percentage: Percentage = (Part / Whole) × 100

Examples:

• 50% means 50 out of 100

• 25% means 25 out of 100

• 75% means 75 out of 100

When we use percentages:

• In school exams (scoring 85% means 85 out of 100)

• In discounts (30% off means we save 30 out of every 100)

• In interest rates (5% interest means 5 rupees per 100 rupees)

• In survey results (statistics)

Question 8: If There are 40 Students in a Class and 32 Pass the Exam, What Percentage Passed?

Solution: Using the percentage formula: Percentage = (Part / Whole) × 100

Part = 32 students (passed) Whole = 40 students (total)

Percentage = (32 / 40) × 100 = 0.8 × 100 = 80%

Answer: 80% of students passed the exam

Question 9: Find 25% of 200

Solution: To find a percentage of a number: Percentage of a number = (Percentage / 100) × Number

25% of 200 = (25 / 100) × 200 = 0.25 × 200 = 50

Answer: 25% of 200 is 50

Another way to think: 25% is one-quarter (1/4), so 1/4 of 200 = 200 ÷ 4 = 50

Question 10: If 15% of a Number is 45, Find the Number

Solution: Let the number be x.

15% of x = 45 (15 / 100) × x = 45 0.15x = 45 x = 45 ÷ 0.15 x = 300

Answer: The number is 300

Verification: 15% of 300 = (15/100) × 300 = 45

Question 11: Convert the Ratio 3:4 to Percentage

Solution: A ratio 3:4 means 3 out of 7 (because 3 + 4 = 7)

First part as percentage = (3 / 7) × 100 = (3 ÷ 7) × 100 = 0.4286 × 100 = 42.86%

Second part as percentage = (4 / 7) × 100 = (4 ÷ 7) × 100 = 0.5714 × 100 = 57.14%

Answer: 3:4 = 42.86% : 57.14% (approximately)

Question 12: Increase 200 by 20%

Solution: Method 1: Find 20% and add 20% of 200 = (20/100) × 200 = 40 Increased value = 200 + 40 = 240

Method 2: Direct calculation If we increase by 20%, the new value is 120% of original 120% of 200 = (120/100) × 200 = 1.2 × 200 = 240

Answer: 200 increased by 20% is 240

Exercise 8.3: Discount and Simple Interest

Question 13: What is a Discount? How Do We Calculate It?

Answer: A discount is a reduction in the original price of an item. When we buy things on sale, the shop gives us a discount.

Discount terminology:

• Marked Price (MP): The price written on the product

• Discount: The amount reduced from the price

• Selling Price (SP): The price we actually pay after discount

How to calculate discount: Discount = (Discount % / 100) × Marked Price

Selling Price = Marked Price - Discount

Or directly: Selling Price = Marked Price × (100 - Discount%) / 100

Example: If a shirt has marked price Rs. 500 and discount is 20%:

• Discount = (20/100) × 500 = Rs. 100

• Selling Price = 500 - 100 = Rs. 400

Question 14: A Book is Marked at Rs. 300 and Sold at a Discount of 15%. What is the Selling Price?

Solution: Marked Price (MP) = Rs. 300 Discount = 15%

Method 1: Calculate discount first Discount = (15 / 100) × 300 = 0.15 × 300 = Rs. 45

Selling Price = 300 - 45 = Rs. 255

Method 2: Direct calculation Selling Price = 300 × (100 - 15) / 100 = 300 × 85 / 100 = 25,500 / 100 = Rs. 255

Answer: The selling price is Rs. 255

Question 15: A Shopkeeper Buys an Item for Rs. 200 and Sells for Rs. 250. What is the Profit?

Solution: Cost Price (CP) = Rs. 200 Selling Price (SP) = Rs. 250

Since SP > CP, there is profit.

Profit = Selling Price - Cost Price = 250 - 200 = Rs. 50

Profit percentage = (Profit / Cost Price) × 100 = (50 / 200) × 100 = 0.25 × 100 = 25%

Answer: Profit = Rs. 50 and Profit % = 25%

Question 16: A Shopkeeper Buys an Item for Rs. 500 and Sells for Rs. 450. What is the Loss?

Solution: Cost Price (CP) = Rs. 500 Selling Price (SP) = Rs. 450

Since SP < CP, there is a loss.

Loss = Cost Price - Selling Price = 500 - 450 = Rs. 50

Loss percentage = (Loss / Cost Price) × 100 = (50 / 500) × 100 = 0.1 × 100 = 10%

Answer: Loss = Rs. 50 and Loss % = 10%

Question 17: What is Simple Interest? How Do We Calculate It?

Answer: Simple interest is the extra money earned when we deposit money in a bank or loan money. The interest is calculated only on the original amount (principal).

Simple Interest terminology:

• Principal (P): The original amount of money

• Rate (R): The percentage of interest per year

• Time (T): The period for which money is kept (in years)

• Simple Interest (SI): The extra money earned

Formula for Simple Interest: Simple Interest = (Principal × Rate × Time) / 100

SI = (P × R × T) / 100

Example: If we put Rs. 1000 in a bank at 5% interest for 2 years: SI = (1000 × 5 × 2) / 100 = Rs. 100

Amount = Principal + SI = 1000 + 100 = Rs. 1100

Question 18: A Person Deposits Rs. 5000 in a Bank at 8% Simple Interest for 3 Years. How Much Interest Will They Get?

Solution: Principal (P) = Rs. 5000 Rate (R) = 8% per year Time (T) = 3 years

Using the formula: Simple Interest = (P × R × T) / 100

SI = (5000 × 8 × 3) / 100 = (5000 × 24) / 100 = 120,000 / 100 = Rs. 1200

Total Amount = Principal + SI = 5000 + 1200 = Rs. 6200

Answer: Simple Interest = Rs. 1200 and Total Amount = Rs. 6200

Question 19: At What Rate of Simple Interest Will Rs. 1000 Become Rs. 1200 in 2 Years?

Solution: Principal (P) = Rs. 1000 Amount = Rs. 1200 Time (T) = 2 years Rate (R) = ? (to find)

Simple Interest = Amount - Principal SI = 1200 - 1000 = Rs. 200

Using SI formula: SI = (P × R × T) / 100 200 = (1000 × R × 2) / 100 200 = (2000 × R) / 100 200 = 20 × R R = 200 ÷ 20 R = 10%

Answer: The rate of simple interest is 10% per year

Verification: SI = (1000 × 10 × 2) / 100 = 200

Question 20: Find the Time Period If Principal is Rs. 2000, Rate is 5% and Simple Interest is Rs. 500

Solution: Principal (P) = Rs. 2000 Rate (R) = 5% per year Simple Interest (SI) = Rs. 500 Time (T) = ? (to find)

Using SI formula: SI = (P × R × T) / 100 500 = (2000 × 5 × T) / 100 500 = (10,000 × T) / 100 500 = 100 × T T = 500 ÷ 100 T = 5 years

Answer: The time period is 5 years

Verification: SI = (2000 × 5 × 5) / 100 = 50,000 / 100 = 500

Mixed Exercise Problems

Question 21: Compare Two Offers and Find Which is Better

A shop offers two ways to buy a TV:

• Offer 1: 20% discount on marked price of Rs. 10,000

• Offer 2: Pay Rs. 8,200

Which offer is better?

Solution: Offer 1: Calculate selling price after 20% discount Discount = (20/100) × 10,000 = Rs. 2,000 Selling Price = 10,000 - 2,000 = Rs. 8,000

Offer 2: Selling Price = Rs. 8,200

Comparison:

• Offer 1: Pay Rs. 8,000 (cheaper)

• Offer 2: Pay Rs. 8,200

Answer: Offer 1 is better as we pay Rs. 8,000 instead of Rs. 8,200, saving Rs. 200

Question 22: A Shopkeeper Gets 10% Discount on Cost Price. If He Sells at Cost Price, What is His Profit or Loss?

Solution: Let original cost price = Rs. 100 (assumed)

Discount received by shopkeeper = 10% of 100 = Rs. 10

Actual cost price for shopkeeper = 100 - 10 = Rs. 90 Selling price = Rs. 100 (sells at original cost price)

Since SP (100) > CP (90), there is profit.

Profit = 100 - 90 = Rs. 10 Profit % = (10/90) × 100 = 11.11%

Answer: The shopkeeper makes a profit of 11.11%

Question 23: Find the Percentage Increase from 80 to 100

Solution: Original value = 80 New value = 100

Increase = 100 - 80 = 20

Percentage increase = (Increase / Original) × 100 = (20 / 80) × 100 = 0.25 × 100 = 25%

Answer: The percentage increase from 80 to 100 is 25%

Question 24: Find the Percentage Decrease from 200 to 150

Solution: Original value = 200 New value = 150

Decrease = 200 - 150 = 50

Percentage decrease = (Decrease / Original) × 100 = (50 / 200) × 100 = 0.25 × 100 = 25%

Answer: The percentage decrease from 200 to 150 is 25%

Question 25: A Dress is Sold for Rs. 2400 at a Profit of 20%. What was the Cost Price?

Solution: Selling Price (SP) = Rs. 2400 Profit = 20%

We know: SP = CP × (100 + Profit%) / 100

2400 = CP × (100 + 20) / 100 2400 = CP × 120 / 100 2400 = CP × 1.2 CP = 2400 ÷ 1.2 CP = Rs. 2000

Answer: The cost price was Rs. 2000

Verification: CP = 2000, Profit = 20% of 2000 = 400, SP = 2000 + 400 = 2400

Tips for Understanding Comparing Quantities

Before solving any problem:

• Identify the two quantities being compared

• Make sure they're in the same units

• Understand which is the base (denominator) for comparison

Read the problem and identify:

• Is this a ratio problem?

• Is this a percentage problem?

• Is this a discount problem?

• Is this a profit/loss problem?

• Is this a simple interest problem?

Once you know the type, use the appropriate formula and method.

Most Repeated Board Questions on Comparing Quantities

Question 1: Simplifying Ratios

Simplify the ratio 36:48

Solution Approach:

1. Find GCD of 36 and 48 = 12

2. Divide both by 12

3. Answer: 3:4

Question 2: Percentage Calculation

Find 30% of 500

Solution Approach:

1. Use formula: (30/100) × 500

2. = 0.3 × 500

3. = 150

Question 3: Discount Problems

A shirt marked Rs. 800 is sold at 25% discount. Find SP

Solution Approach:

1. Calculate discount: (25/100) × 800 = 200

2. SP = 800 - 200 = 600

3. Or: 800 × 75/100 = 600

Question Type 4: Profit and Loss

CP is Rs. 400, SP is Rs. 480. Find profit %

Solution Approach:

1. Profit = 480 - 400 = 80

2. Profit % = (80/400) × 100 = 20%

Question 5: Simple Interest

Find SI on Rs. 2000 at 10% for 3 years

Solution Approach:

1. SI = (2000 × 10 × 3) / 100

2. = 60,000 / 100

3. = Rs. 600

Question 6: Percentage Increase/Decrease

Increase 150 by 20%

Solution Approach:

1. 20% of 150 = 30

2. Result = 150 + 30 = 180

3. Or: 150 × 120/100 = 180

Question 7: Finding Original Value from Percentage

20% of a number is 60. Find the number

Solution Approach:

1. Let number = x

2. (20/100) × x = 60

3. x = 60 × 100/20 = 300

Question 8: Comparing Quantities Using Ratios

The ratio of boys to girls is 3:2. If there are 15 boys, how many girls?

Solution Approach:

1. If boys = 3k, girls = 2k

2. 3k = 15, so k = 5

3. Girls = 2 × 5 = 10

Question 9: Reverse Profit/Loss Calculation

An item is sold at Rs. 540 with 10% profit. Find CP

Solution Approach:

1. SP = CP × 110/100

2. 540 = CP × 1.1

3. CP = 540/1.1 = Rs. 490.91

Question 10: Combined Discounts

Two successive discounts of 10% and 20% are given. Find net discount %

Solution Approach:

1. After 10% discount: price becomes 90% of original = 0.9P

2. After 20% on this: 0.8 × 0.9P = 0.72P

3. Net discount = 28%

Most Common Examination Questions (Board Exams)

1-2 Mark Questions (Very Frequently Asked):

1. Write the ratio 2:5 in the form of percentage Answer: 2/(2+5) × 100 = 28.57% and 5/(2+5) × 100 = 71.43%

2. Find 50% of 200 Answer: (50/100) × 200 = 100

3. If CP is 500 and SP is 600, find profit % Answer: (100/500) × 100 = 20%

4. A book costs Rs. 300 with 10% discount. What do we pay? Answer: 300 × 90/100 = Rs. 270

5. Find simple interest on Rs. 1000 at 5% for 2 years Answer: (1000 × 5 × 2) / 100 = Rs. 100

6. If 15% of x is 45, find x Answer: x = 45 × 100/15 = 300

7. Simplify ratio 12:18 Answer: 2:3

8. Find 25% of 400 Answer: 100

9. What is marked price if SP is 720 after 20% discount? Answer: 720 × 100/80 = 900

10. If profit is 25%, and CP is 200, find SP Answer: 200 × 125/100 = 250

3-4 Mark Questions (Frequently Asked):

1. A shopkeeper buys at Rs. 400 and sells at Rs. 500. Find profit % Solution: Profit = 100, Profit % = (100/400) × 100 = 25%

2. Three quantities are in ratio 2:3:5. If smallest is 20, find others Solution: 2k=20, k=10; then 3k=30, 5k=50

3. Find discount % if MP is 1000 and SP is 800 Solution: Discount = 200, Discount % = (200/1000) × 100 = 20%

4. A man gets 15% increase in salary from 20000. Find new salary Solution: 20000 × 115/100 = Rs. 23000

5. Find SI on Rs. 5000 at 8% for 2.5 years Solution: (5000 × 8 × 2.5) / 100 = Rs. 1000

6. If a number increases by 20% to become 240, find original number Solution: Original × 120/100 = 240, Original = 200

7. Two quantities in ratio 3:4. If sum is 350, find individual quantities Solution: 3x + 4x = 350, x = 50; quantities are 150 and 200

8. Find rate if principal Rs. 2000 becomes Rs. 2400 in 2 years at SI Solution: SI = 400, (2000 × R × 2) / 100 = 400, R = 10%

9. A dress costs Rs. 1500 with two discounts of 10% and 5% Solution: After 10%: 1350; After 5% on 1350: 1282.50

10. Find cost price if selling price is Rs. 880 with 12% loss Solution: SP = CP × 88/100, 880 = CP × 0.88, CP = 1000

5-6 Mark Questions (Less Frequent but Important):

1. Compare two investment schemes: 1) Rs. 10000 at 8% for 3 years, 2) Rs. 10000 at 7% for 4 years Solution: Calculate SI for both and compare: 2400 vs 2800

2. A shop gives 20% discount and makes 10% profit. If CP is Rs. 500, find MP Solution: SP = CP × 110/100 = 550; MP × 80/100 = 550; MP = 687.5

3. The population increases from 80000 to 96000 in 2 years. Find annual percentage increase Solution: 80000 × (1 + r/100)² = 96000; r = 10% (approximately)

4. Three books cost Rs. 200, 300, and 400. Discounts are 10%, 15%, and 20% respectively. Find total SP Solution: 180 + 255 + 320 = 755

5. A shopkeeper sells 10 items at Rs. 500 each with 20% profit. Find CP of all items Solution: SP = 5000; CP = 5000 × 100/120 = 4166.67

6. Simple interest on Rs. 5000 for 3 years is same as SI on Rs. 3000 for 5 years. Find rate Solution: (5000 × R × 3) / 100 = (3000 × R × 5) / 100; This shows rates must be different

7. A quantity increases by 25% then decreases by 20%. Find net percentage change Solution: Original × 1.25 × 0.8 = original × 1.0 = no change

8. Ratio of income to expenditure is 5:3. If income increases by 20% and expenditure by 10%, find new ratio Solution: New income = 5 × 1.2 = 6; New expenditure = 3 × 1.1 = 3.3; New ratio = 60:33 = 20:11

9. A man buys 50 items at Rs. 10 each, sells 40 at Rs. 15 each and 10 at Rs. 8 each. Find profit/loss % Solution: CP = 500; SP = 600 + 80 = 680; Profit = 180; Profit % = 36%

10. Two successive discounts are equivalent to single discount of 28%. If one is 20%, find the other Solution: 100 × 0.8 × (1-d) = 72; 1-d = 0.9; d = 10%