Percentage Exercises with Answers and Word Problems

Percentages are a fundamental concept in mathematics that are used to express numbers as parts of 100. They are widely applied in everyday situations such as calculating discounts, interest rates, profit and loss, exam scores, etc. Understanding percentages helps build a strong foundation for topics like ratios, fractions, and algebra. This page provides a comprehensive set of percentage exercises designed to improve your problem-solving skills from basic calculations to real-life word problems. 

Table of Contents

• What Is a Percentage?
• Key Formulas
• The Fraction-Percentage Equivalents
• Exercise Set 1: Basic Percentage Calculations
• Exercise Set 2: Finding the Whole from a Percentage
• Exercise Set 3: Percentage Increase and Decrease
• Exercise Set 4: Percentage in Profit, Loss, and Discounts
• Exercise Set 5: Depreciation and Growth
• Exercise Set 6: Real-Life and Word Problems
• Common Mistakes to Avoid

What Is a Percentage?

The word ‘percentage’ comes from the Latin 'per centum', which literally means ‘by the hundred’. A percentage is simply a fraction with a denominator of 100.

So when we say 45%, we mean 45 out of every 100, or 45/100, or the decimal 0.45.

Basic conversion facts:

50% = 50/100 = 1/2 = 0.50

25% = 25/100 = 1/4 = 0.25

75% = 75/100 = 3/4 = 0.75

10% = 10/100 = 1/10 = 0.10

1% = 1/100 = 0.01

100% = the whole thing

The standard formula is:

Percentage = (Value / Total Value) × 100

Value = (Percentage / 100) × Total

Key Formulas

The Fraction-Percentage Equivalents

Memorising these fraction percentage equivalents saves enormous time in exams:

Exercise Set 1: Basic Percentage Calculations

Exercise 1.1: Find 35% of 280.

Solution: 35% of 280 = (35/100) × 280 = 98

Exercise 1.2: What percentage of 150 is 45?

Solution:  x% = (45/150) × 100 = 30%

30% of 150 is 45.

Exercise 1.3: 72 is 40% of what number?

Solution: Let the number be x.

40% of x = 72

(40/100) × x = 72

x = (72 × 100)/40 = 180

The required number is 180

Exercise 1.4: Convert 3/8 to a percentage.

Solution: 3/8 × 100 = 37.5%

Exercise 1.5: There are 50 students in a class. If 14% are absent on a particular day, how many students are present?

Solution: Total number of students = 50

Students absent = 14% of 50 = (14/100) × 50 = 7

Students present = 50 − 7 = 43

There are 43 students present.

Exercise 1.6: In a basket of apples, 12% are rotten and 66 are in good condition. Find the total number of apples.

Solution: Good apples = 100% − 12% = 88% of total

88% of x = 66

x = (66 × 100)/88 = 75

The total number of apples is 75.

Exercise 1.7: Express 0.085 as a percentage.

Solution:0.085 × 100 = 8.5%
0.085 as a percentage is 8.5%.

Exercise 1.8: A school has 970 students, of whom 60% are boys. Find the number of girls.

Solution: Boys = 60% of 970 = 582

Girls = 970 − 582 = 388

The number of girls in school is 388.

Exercise Set 2: Finding the Whole from a Percentage

Exercise 2.1: A fruit seller sold 40% of his apples and still has 420 left. How many apples did he originally have?

Solution: Given fruit seller sold 40% of his apples and still has 420 left. 

Remaining = 100% − 40% = 60% of total

60% of x = 420

x = (420 × 100)/60 = 700

There are 700 apples originally.

Exercise 2.2: An alloy contains 26% copper. How much alloy is needed to obtain 260 g of copper?

Solution: Let the required amount of alloy be x.

26% of x = 260

x = (260 × 100)/26 = 1000 g

1000 g of alloy is required to get 260 g of copper.

Exercise 2.3: A defect-finding machine rejects 0.085% of all cricket bats. On a particular day it rejected 34 bats. How many bats were manufactured?

Solution: Let n be the number of bats manufactured.

0.085% of n = 34

n = (34 × 100)/0.085 = 40,000

40,000 bats are manufactured.

Exercise 2.4: For a student to pass an examination, he must score 55%. He gets 120 marks and fails by 78 marks. What are the total marks?

Solution: Pass percentage = 55%

Passing marks = 120 + 78 = 198

55% of total = 198

Total = (198 × 100)/55 = 360

Total marks = 360

Exercise 2.5: 25% of a number is 8 less than one-third of that number. Find the number.

Solution: Let n be the required number.

(n/3) − (25n/100) = 8

(n/3) − (n/4) = 8

(4n − 3n)/12 = 8

n/12 = 8

n = 96

Exercise Set 3: Percentage Increase and Decrease

Exercise 3.1: A shirt costs ₹800. Its price is increased by 15%. What is the new price?

Solution: Given cost of shirt =  ₹800.

Increase = 15% of 800 = ₹120

New price = 800 + 120 = ₹920

Exercise 3.2: The price of petrol dropped from ₹95 per litre to ₹76. What is the percentage decrease?

Solution: Decrease in petrol price = 95 − 76 = 19

% Decrease = (19/95) × 100 = 20%

Therefore, there is a 20% decrease in petrol prices.

Exercise 3.3: A number is increased by 20% and then decreased by 20%. What is the net change?

Solution: Let the number be 100.

After 20% increase, it equals 120

After 20% decrease = 120 × (80/100) = 96

Net change = 100 − 96 = 4 (decrease)

Net % change = 4% decrease

Exercise 3.4: The price of a product is first decreased by 25% and then increased by 20%. What is the net percentage change?

Solution: Let original price = ₹100

After 25% decrease, the price = ₹75

After 20% increase, the price = 75 × (120/100) = ₹90

Net change = ₹10 decrease

Net % change = 10% decrease

Exercise 3.5: A number is decreased by 10% and then increased by 10%. The final number is 10 less than the original. Find the original number.

Solution: Let the original number be x.

110% of (90% of x) = (99/100)x

Given the final number is 10 less than the original

⇒ x - (99/100)x = 10

x/100 = 10

x = 1000

Exercise Set 4: Percentage in Profit, Loss, and Discounts

Exercise 4.1: A shopkeeper buys a watch for ₹1,200 and sells it for ₹1,500. Find the profit percentage.

Solution: Cost price = CP =  ₹1,200

Selling price = SP = ₹1,500. 

Profit = 1500 − 1200 = ₹300

Profit % = (Profit/CP) × 100  = (300/1200) × 100 = 25%

Exercise 4.2: A toy is marked at ₹600 and sold at a discount of 15%. Find the selling price.

Solution: Marked price = ₹600

Discount = 15% of 600 = ₹90

Selling price = 600 − 90 = ₹510

Exercise 4.3: An article is sold at a loss of 12%. If the cost price is ₹2,500, find the selling price.

Solution: Cost price = CP =  ₹2,500

Selling price = 100% − 12% = 88% of CP

SP = (88/100) × 2500 = ₹2,200

Therefore, the selling price of the article is ₹2,200

Exercise 4.4: A dealer sells goods at a 10% profit after giving a 20% discount on the marked price. Find the ratio of cost price to marked price.

Solution: Let marked price be 100.

Selling price after 20% discount = 80

SP = 110% of CP (since 10% profit)

CP = (80 × 100)/110 = 72.72...

CP:MP = 72.72:100 ≈ 8:11

Exercise 4.5: A person buys two items for ₹2,000 each. He sells one at 25% profit and the other at 25% loss. What is his overall gain or loss percentage?

Solution:

Total cost = ₹4,000

SP of first = (125/100) × 2000 = ₹2,500

SP of second = 2000 × 0.75 = ₹1,500

Total SP = ₹4,000

SP - CP = 0

Therefore, there is no gain or loss.

Exercise Set 5: Depreciation and Growth

Exercise 5.1: A washing machine currently costs ₹8,748. Its value depreciates at 10% per year. What was its price 3 years ago?

Solution: Current value = Original × (1 − r/100)^n

8748 = Original × (90/100)³

8748 = Original × (729/1000)

Original = 8748 × (1000/729) = ₹12,000

The original price of washing machine is ₹12,000

Exercise 5.2: A city's population is 200,000 and grows at 5% per year. What will it be after 2 years?

Solution: Final population = Initial population × (1 + r/100)^n

Population after 2 years = 200000 × (1 + 5/100)²

= 200000 × (1.05)²

= 200000 × 1.1025

= 220,500

Population after 2 years is 220,500.

Exercise 5.3: A car worth ₹500,000 depreciates at 8% per year. Find its value after 2 years.

Solution: Value after 2 years = 500000 × (1 − 8/100)²

= 500000 × (0.92)²

= 500000 × 0.8464

= ₹4,23,200

The value of the car after two years is ₹4,23,200.

Exercise Set 6: Real-Life and Word Problems

Exercise 6.1: A person multiplied a number by 3/5 instead of 5/3. What is the percentage error in the calculation?

Solution: Let the number be x.

Correct result = 5x/3

Incorrect result = 3x/5

Error = 5x/3 − 3x/5 = (25x − 9x)/15 = 16x/15

% Error = (16x/15) / (5x/3) × 100 = (16/15 × 3/5) × 100 = 64%

Exercise 6.2: If 20% of x = y, what is the value of y% of 20 (in terms of x)?

Solution:

20% of x = y ⇒  y = x/5

y% of 20 = (y/100) × 20 = (x/5 ÷ 100) × 20 = (x × 20)/(500) = x/25 = 4% of x

Therefore, y% of 20 = 4% of x

Exercise 6.3: The difference between two numbers x and y (x > y) is 100. Also, 10% of x equals 15% of y. Find both numbers.

Solution: 10% of x = 15% of y

10x = 15y 

⇒ x = 1.5y

x − y = 100 

⇒ 1.5y − y = 100 

⇒  0.5y = 100 

⇒  y = 200

⇒ x = 1.5y = 1.5 × 200 = 300

Therefore, x = 300 and y = 200

Exercise 6.4: A man gave 40% of his retirement money to his wife. He then gave 20% of the remaining amount to each of his 3 sons. 50% of what was left was spent on miscellaneous items. The remaining ₹1,20,000 was deposited in the bank. How much did he receive as retirement money?

Solution: Let retirement money be ₹100n.

Money given to wife = 40n 

⇒ remaining money = 60n

Money given to 3 sons = 3 × 20% of 60n = 36n 

⇒ remaining = 60n -  36n = 24n

Miscellaneous = 50% of 24n = 12n 

⇒remaining = 12n

12n = 1,20,000 

⇒ n = 10,000

Total money = 100 × 10,000 = ₹10,00,000

Therefore, he received ₹10,00,000 as retirement money.

Exercise 6.5: In a gaming event, 75% of registered participants turned up. Of those, 2% were declared unfit. The winner defeated 9,261 valid participants, which was 75% of total valid participants. How many people registered?

Solution: Let registered number of participants = n

Number of participants who turned up = 0.75n

Invalid participants = 2%

Valid = 98% of 0.75n = 0.735n

Winner defeated 75% of valid = 0.75 × 0.735n = 0.55125n = 9261

0.55125n = 9261

⇒ n = 9261/0.55125 = 16,800

Therefore, there are 16,800 registered participants.

Exercise 6.6: A broker charges 5% commission on orders up to ₹10,000 and 4% on amounts above that. He remits ₹31,100 to his client after deducting commission. Find the total order amount.

Solution: Let the total order be ₹n.

Commission on first ₹10,000 = 5% of 10,000 = ₹500

Let amount above ₹10,000 = ₹(n − 10,000)

Commission above = 4% of (n − 10,000)

Total remitted = n − 500 − 0.04(n − 10,000) = 31,100

⇒ n − 500 − 0.04n + 400 = 31,100

⇒ 0.96n − 100 = 31,100

⇒ 0.96n = 31,200

⇒ n = 32,500

Therefore, total order = ₹32,500

Common Mistakes to Avoid

• Applying percentages on the wrong base: A price increased by 20% and then decreased by 20% does NOT return to the original. The second percentage is always applied on the new value, not the original.

• Adding successive percentages directly: Two successive increases of 10% are NOT the same as a single 20% increase. The correct answer is 21%. Always use the formula a + b + (ab/100).

• Confusing % of A out of B vs % A is of B: What % is 30 of 150? =  (30/150) × 100 = 20%; Find 30% of 150 = (30/100) × 150 = 45. These are completely different operations.

•  Forgetting to work on the original (base) value: For percentage error or profit/loss, always use the original or true value as the denominator, not the final or incorrect value.

• Assuming that profit% on SP = profit% on CP: Profit percentage is always calculated on the cost price. Discount percentage is on the marked price. Keep the bases straight.