MathsCBSE Class 8 · Fractions in Disguise

Percentages and Percentage Change for Class 8

Updated 17 October 2026 · Fact-checked

A percentage is a fraction with denominator 100: y% of a quantity Q is (y/100) x Q. To convert a fraction to a percentage multiply by 100; to convert a percentage to a fraction divide by 100 and simplify. Percentages of different wholes cannot be compared directly. A percentage change is (change / original) x 100, and successive changes multiply: a rise of 20% then a fall of 20% leaves a 4% fall.

The 20% fall is taken on the larger 120, not on 100, so the final 96 ends up 4% below the start.

Illustration drawn for QuestionBuddy.

Understand Percentages and Percentage Change

Meaning. The symbol % is read "per cent", from the Latin *per centum*, "by the hundred". So $25% = \frac{25}{100} = \frac{1}{4}$ : 25 out of every 100. The chapter notes that the idea of "per hundred" appears as early as the 4th century BCE in Kautilya's text on interest rates.

Fraction to percentage. A fraction is "of a unit"; a percentage is "per 100". Multiply the fraction by 100: $\frac{7}{8} = \frac{7}{8} \times 100% = 87.5%$ . Or write an equivalent fraction with denominator 100: $\frac{3}{4} = \frac{75}{100} = 75%$ . If 15 of 25 marbles are white, the percentage is $\frac{15}{25} \times 100 = 60%$ .

Percentage to fraction. Divide by 100 and simplify: $24% = \frac{24}{100} = \frac{6}{25}$ ; $35% = \frac{7}{20}$ .

Why percentages? They put different fractions on a common scale of 100 (round, easy to picture in a base-10 number system), so comparing 3/4 with 4/5 becomes comparing 75% with 80%. Other "per" scales also appear, such as per thousand or per lakh.

Percentage of a quantity. $y %$ of $Q$ is $\frac{y}{100} \times Q$ . For example, 25% of 120 g is $30$ g. Useful benchmarks: 50% is half, 25% is a quarter, 10% is one tenth, and 20% is double 10%.

Compare only the same whole. Madhu's biscuits have 25% sugar and Madhav's have 35%. You cannot say who ate more sugar until you know the weights eaten: 25% of 120 g is 30 g, while 35% of 95 g is 33.25 g, so Madhav ate more.

Percentage change. $ $percentage change = \frac{new - old}{old} \times 100%$ $p os i t i v e f or anin cr e a se an d n e g a t i v e f or a d ecr e a se . T h e ba se i s a l w a y s t h e * * o l d * * v a l u e, so a r i se f r o m 240 t o 294 i s$ 22.5\% $b u t a f a l l f r o m 294 t o 240 i so n l y ab o u t$ 18.4\%$.

Multipliers. An increase of $y %$ multiplies by $1 + \frac{y}{100}$ and a decrease multiplies by $1 - \frac{y}{100}$ . Successive changes multiply the multipliers: a 20% rise then a 20% fall is $1.2 \times 0.8 = 0.96$ , a 4% net fall. A fall of 20% then a rise of 25% is $0.8 \times 1.25 = 1$ , no net change.

Profit, loss, discount, tax and interest are on the next topic page of this chapter.

Percentages and Percentage Change: formulas to remember

Percentage of a quantity: $y % of Q = \frac{y}{100} \times Q$; Convert y% to y/100.
Fraction to percentage: $\frac{a}{b} = \frac{a}{b} \times 100%$; Multiply by 100.
Percentage change: $\frac{new - old}{old} \times 100%$; The base is always the old value.
Multiplier for a change: $increase y % : 1 + \frac{y}{100}; decrease y % : 1 - \frac{y}{100}$; Successive changes multiply.
Successive changes: $final = original \times m_{1} \times m_{2}$; m1 and m2 are the multipliers.

How to solve Percentages and Percentage Change questions

Use this routine for percentage questions.

1Decide what the whole (the 100%) is in the question.
2To find a percentage of a quantity, convert the percentage to a fraction and multiply.
3To find what percentage one quantity is of another, divide and multiply by 100.
4For a change, subtract old from new, divide by the old, and multiply by 100.
5For a given part and its percentage, find the whole by dividing the part by the percentage fraction.
6For several changes, convert each to a multiplier and multiply them.
7Check that the answer is reasonable (for example 50% should be about half).

Percentages and Percentage Change in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Percentages of different wholes

Suppose Ria drinks 250 mL of juice with 18% pulp while Sam drinks 180 mL of juice with 26% pulp.

Who drinks more pulp, and by how much?

Try it first, then show the solutionSolution

Ria: $0.18 \times 250 = 45$ mL.
Sam: $0.26 \times 180 = 46.8$ mL.

Answer: Sam drinks more pulp, by 1.8 mL.

Memory hook: A higher percentage of a smaller whole can still be a bigger amount.

Percentage change

Imagine a town whose population grows from 2,40,000 to 2,94,000.

Find the percentage increase.

Try it first, then show the solutionSolution

Increase $= 294000 - 240000 = 54000$ .
Percentage increase $= \frac{54000}{240000} \times 100 = 22.5%$ .

Answer: 22.5%

Memory hook: Always divide the change by the old value.

Tricks to remember Percentages and Percentage Change

Short memory aids. Say each one aloud once, then use it on a question the same day.

Per cent means over 100
x% is x over 100; 25% is one quarter, 50% half, 75% three quarters.
Use it when: Quick conversions.
Same whole or no comparison
Percentages from different wholes cannot be compared until they are turned into actual amounts.
Use it when: Madhu and Madhav type questions.
Old is the base
Percentage change uses the old value as the base; the reverse change has a different percentage.
Use it when: Rise and fall questions.
Up 20 then down 20 is not zero
Multiply 1.2 by 0.8 and you get 0.96, a 4% fall.
Use it when: Successive change questions.

Percentages and Percentage Change: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Percentage as a fraction · 1 mark · MCQBoard level

35% written as a fraction in simplest form is: (a) $\frac{7}{20}$ (b) $\frac{35}{10}$ (c) $\frac{3}{5}$ (d) $\frac{7}{10}$

Try it first, then show the solutionSolution

$35% = \frac{35}{100}$ , and dividing numerator and denominator by 5 gives $\frac{7}{20}$ .

Answer: (a) 7/20

Example 2: Fractions to percentages · 2 marks · Short answerBoard level

Express as percentages: (i) 7/8 (ii) 18 out of 40.

Try it first, then show the solutionSolution

(i) $\frac{7}{8} \times 100 = 87.5%$ . (1 mark)
(ii) $\frac{18}{40} \times 100 = 45%$ . (1 mark)

Answer: (i) 87.5% (ii) 45%

Example 3: Different wholes · 3 marks · Short answerHarder board level

Ria drinks 250 mL of juice with 18% pulp and Sam drinks 180 mL of juice with 26% pulp. Who drinks more pulp and by how much?

Try it first, then show the solutionSolution

Ria's pulp: $\frac{18}{100} \times 250 = 45$ mL. (1 mark)
Sam's pulp: $\frac{26}{100} \times 180 = 46.8$ mL. (1 mark)
Sam drinks more, by $46.8 - 45 = 1.8$ mL, although Ria drank more juice. (1 mark)

Answer: Sam, by 1.8 mL

Example 4: Increase and decrease between the same two numbers · 3 marks · Short answerHarder board level

A town grows from 2,40,000 to 2,94,000 people. Find the percentage increase, and the percentage decrease if it fell back from 2,94,000 to 2,40,000 (to one decimal place). Why are they different?

Try it first, then show the solutionSolution

Increase $= 54000$ ; $\frac{54000}{240000} \times 100 = 22.5%$ . (1 mark)
Decrease $= 54000$ ; $\frac{54000}{294000} \times 100 \approx 18.4%$ . (1 mark)
The same change of 54,000 is divided by different old values (2,40,000 and 2,94,000), so the percentages differ. (1 mark)

Answer: 22.5% increase; about 18.4% decrease; different bases.

Example 5: Successive changes · 4 marks · Long answerHarder board level

A price of Rs 500 is raised by 20% and then the new price is lowered by 20%. Find the final price and the net percentage change. What if the price is first lowered by 20% and then raised by 25%?

Try it first, then show the solutionSolution

After the 20% rise: $500 \times 1.2 = 600$ . (1 mark)
After the 20% fall: $600 \times 0.8 = 480$ , a net fall of $\frac{20}{500} \times 100 = 4%$ . (1 mark)
Lowered by 20% then raised by 25%: $500 \times 0.8 = 400$ , then $400 \times 1.25 = 500$ . (1 mark)
Net change is 0%, since $0.8 \times 1.25 = 1$ . (1 mark)

Answer: Rs 480 and a 4% net fall; in the second case Rs 500 and no net change.

Where students lose marks in Percentages and Percentage Change

The slips examiners see most often, and what to write instead.

Mistake: Using the new value as the base for percentage change
The base is the old (original) value.
Wrong working
"240 to 294: 54 / 294 x 100 = 18.4% increase."
Right working
$\frac{54}{240} \times 100 = 22.5%$ .
Do this: Underline "from" in the question; that value is the base.
Usually costs: The answer mark.
Mistake: Adding successive percentages
Successive changes act on different bases, so they multiply.
Wrong working
"+20% then -20% gives 0% change."
Right working
$1.2 \times 0.8 = 0.96$ , a 4% fall.
Do this: Convert each change to a multiplier.
Usually costs: 2 marks.
Mistake: Comparing percentages of different wholes
Only equal wholes allow a comparison of percentages.
Wrong working
"26% is more than 18%, so Sam drank more pulp."
Right working
$0.26 \times 180 = 46.8$ mL against $0.18 \times 250 = 45$ mL: true here, but not always; compute the amounts.
Do this: Turn each percentage into an amount.
Usually costs: The reasoning mark.
Mistake: Forgetting to multiply by 100
A fraction is a percentage only after multiplying by 100.
Wrong working
"18 out of 40 is 0.45%"
Right working
$0.45 \times 100 = 45%$ .
Do this: Write the % sign only after multiplying by 100.
Usually costs: 1 mark.

Exam tips

Write the base (the whole) in words before computing a percentage.
Use multipliers (1.2, 0.8) for successive changes: they avoid base confusion.
To find the whole from a part, divide the part by the percentage as a decimal.
Check that increases give a result bigger than the original and decreases smaller.
Give units and the percent sign in the final line.

Practice questions on Percentages and Percentage Change

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · Assertion-reasonBoard level
Assertion (A): 25% of 40 equals one fourth of 40.
Reason (R): 25% means 25 out of 100, which is one fourth.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
Convert 25% to a fraction.
Answer
(a). $25% = \frac{25}{100} = \frac{1}{4}$ , so 25% of 40 is $\frac{1}{4} \times 40 = 10$ .
Q2 · 2 marks · Very short answerBoard level
Arrange in ascending order: 3/5, 55%, 0.58.
Hint
Write all as percentages.
Answer
$\frac{3}{5} = 60%$ , $55%$ , $0.58 = 58%$ . Ascending: $55%, 0.58, \frac{3}{5}$ .
Q3 · 3 marks · Short answerHarder board level
A tank is 35% full. After 36 litres are added it is 65% full. Find its capacity and the amount of water initially in it.
Hint
36 litres is the difference of two percentages of the capacity.
Answer
The added 36 L is $65% - 35% = 30%$ of the capacity, so capacity $= 36 \div 0.30 = 120$ L. Initial water $= 35% \times 120 = 42$ L. Check: $42 + 36 = 78 = 65%$ of 120.
Q4 · 4 marks · Long answerHOTS / competency
In a school of 1200 students, 55% are girls. 40% of the girls and 25% of the boys cycle to school. How many students cycle, and what percentage of the whole school is that?
Hint
Find girls and boys first, then the cyclists in each group.
Answer
Girls $= 0.55 \times 1200 = 660$ ; boys $= 1200 - 660 = 540$ . Cycling girls $= 0.4 \times 660 = 264$ ; cycling boys $= 0.25 \times 540 = 135$ . Total $= 264 + 135 = 399$ . Percentage $= \frac{399}{1200} \times 100 = 33.25%$ .
Q5 · 4 marks · Case-basedHOTS / competency
Case study: a town has 8000 people and 62.5% of them own a smartphone.
(i) How many own a smartphone? (1 mark)
(ii) Of these, 30% use a payment app. What percentage of the whole town is this? (1 mark)
(iii) Next year the number of smartphone owners rises by 10% while the town stays at 8000. What percentage of the town owns a smartphone? (1 mark)
(iv) If instead both the town population and the smartphone owners rise by 10%, what happens to the percentage? Explain. (1 mark)
Hint
In (iv), both numbers are multiplied by the same factor.
Answer
(i) $0.625 \times 8000 = 5000$ . (ii) $0.3 \times 5000 = 1500$ users, and $\frac{1500}{8000} \times 100 = 18.75%$ of the town. (iii) $5000 \times 1.1 = 5500$ , and $\frac{5500}{8000} \times 100 = 68.75%$ . (iv) Population becomes 8800 and owners become 5500: $\frac{5500}{8800} \times 100 = 62.5%$ , unchanged, because both numbers are multiplied by the same factor so their ratio stays the same.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.What does per cent mean?Show answer
Out of one hundred (from Latin per centum).
2.How do you convert a fraction to a percentage?Show answer
Multiply by 100.
3.What is the base for percentage change?Show answer
The old (original) value.
4.What multiplier represents a 15% decrease?Show answer
0.85.
5.What is 20% of a number compared with 10%?Show answer
Double.
6.What is the result of a 20% rise followed by a 20% fall?Show answer
A 4% net fall.

Percentages and Percentage Change: frequently asked questions

Can a percentage be more than 100?

Yes. 150% of a quantity is one and a half times the quantity, as when a final value is higher than the starting value.

Why not just use fractions?

Percentages give a common scale of 100 that is easy to compare and visualise, and fit a base-10 number system.

Is a 10% rise followed by a 10% fall the original value?

No: 1.1 x 0.9 = 0.99, a 1% fall.

How do I find 15% quickly?

Find 10%, then add half of that (5%).