Percentage: Find, Compare, Increase, Decrease and Work Backwards

By the end of this lesson you will be able to find any percentage of a number, say what percentage one amount is of another, work out a percentage increase or decrease, combine two changes in a row, and find the original value after a change.

Already confident? Skip to practice: Percentage quiz

Before this: you should be able to turn a fraction into a percentage and back (1/4 = 25%, 3/8 = 37.5%). If that feels shaky, read the Fractions, Decimals and Percentages lesson first.

What "per cent" means

Per cent means "out of 100", and its sign is %. So 7% means 7 out of every 100. As a fraction that is 7/100, and as a decimal it is 0.07.

Some words used below:

• The base (or whole) is the amount you take the percentage of. In "20% of 450", the base is 450.
• The original value is the value before a change. The new value is the value after it.
• A multiplier makes a change in one step. A 15% increase multiplies by 1.15 (that is, 115/100). A 15% decrease multiplies by 0.85 (85/100).

One rule for the whole lesson: a percentage change is worked out on the value just before the change (its original value), unless the question says otherwise. When two changes come one after the other, the second is worked out on the value after the first.

Worked examples

Read each example slowly. The first two show every step. In the later ones, some steps are left for you, with the answer just below.

Example 1: A percentage of a number, and the whole from a part

Question: (a) Find 35% of 240. (b) In a village, 35% of the families keep cows, and that is 84 families. How many families live in the village? How many do not keep cows?

Part (a): 35% of 240

1. "35% of" means "35/100 times". So 35% of 240 = 35/100 × 240.
2. Multiply first: 35 × 240 = 8,400.
3. Then divide by 100: 8,400 ÷ 100 = 84.

A second way, with no long multiplication: build 35% from easy pieces.

1. 10% of 240 is 240 ÷ 10 = 24.
2. So 30% is 3 × 24 = 72.
3. 5% is half of 10%, so 5% is 24 ÷ 2 = 12.
4. 35% = 30% + 5% = 72 + 12 = 84.

Part (b): the whole from a part

1. Write what you know: 35% of the families = 84 families.
2. 35% is 7 lots of 5%, so 5% of the families = 84 ÷ 7 = 12 families.
3. 100% is 20 lots of 5%, so all the families = 12 × 20 = 240 families.
4. The same in one line: whole = part × 100 ÷ percentage = 84 × 100 ÷ 35 = 8,400 ÷ 35 = 240.
5. The families without cows are the rest: 100% − 35% = 65% of 240. That is 65/100 × 240 = 156 families. Check: 84 + 156 = 240.

The principle: "x% of N" means x/100 × N. When you know the part and its percentage, find a small piece first (1%, or 5%, or 10%), then build up to 100%. The rest of a whole is 100% minus the part.

Example 2: One amount as a percentage of another

Question: In a mock test, Priya scored 54 out of 75. In a different test, her brother Aman scored 63 out of 90. Write each score as a percentage. Who did better? Then: what percentage of ₹2 is 50 paise?

1. A percentage of a total = (the part ÷ the total) × 100. The total goes on the bottom.
2. Priya: 54/75 × 100. Simplify first: divide the top and the bottom by 3 to get 18/25. Then 18/25 × 100 = 18 × 4 = 72%.
3. Aman: 63/90 × 100. Divide the top and the bottom by 9 to get 7/10. Then 7/10 × 100 = 70%.
4. 72% is more than 70%, so Priya did better, even though Aman scored more marks. Percentages compare the two scores as if both tests were out of 100.
5. For the money question, put both amounts in the same unit first: ₹2 = 200 paise.
6. 50/200 × 100 = 25%.

The principle: "A as a percentage of B" is A ÷ B × 100. The amount after "of" (here, B) goes on the bottom. Both amounts must be in the same unit.

Example 3: Percentage increase and decrease

Question: The monthly rent of a flat goes up from ₹8,000 to ₹9,000. By what percentage did it rise? A year later it comes back down from ₹9,000 to ₹8,000. By what percentage did it fall?

1. The rise is 9,000 − 8,000 = ₹1,000.
2. The value before the rise is ₹8,000, so ₹8,000 is the base.
3. Percentage increase = 1,000/8,000 × 100 = 1/8 × 100 = 12.5%.
4. For the fall, the change is again ₹1,000. But the value before the fall is ₹9,000, so now ₹9,000 is the base.
5. Your turn: work out the percentage decrease.

Check: 1,000/9,000 × 100 = 1/9 × 100 = $11\tfrac{1}{9}\%$ (about 11.11%). The same ₹1,000 is a smaller percentage of the bigger base.

The principle: percentage change = (the change ÷ the original value) × 100. The base is the value before the change.

To make a change in one step, use a multiplier. An increase of r% multiplies by (100 + r)/100, and a decrease of r% multiplies by (100 − r)/100. For example, ₹8,000 increased by 12.5% is 8,000 × 112.5/100 = 8,000 × 1.125 = ₹9,000.

Two kinds of question use this same rule:

• "More than" and "less than". In a comparison, the base is the amount after "than". Raju earns ₹9,000 and Shyam earns ₹8,000. Raju earns 12.5% more than Shyam, because the base is Shyam's pay. But Shyam earns $11\tfrac{1}{9}\%$ less than Raju, because now the base is Raju's pay. In general, if A is r% more than B, then B is r × 100 ÷ (100 + r) per cent less than A.
• Price and quantity. Cooking oil goes up from ₹150 to ₹180 a litre, a rise of 20%. A family wants to spend the same amount on oil as before. Spending = price × quantity, so the quantity must shrink to 150/180 = 5/6 of what it was. That is a cut of 1/6, or 16⅔%, which is less than the 20% rise in price. The cut is the price rise divided by the new price: 30/180 = 1/6.

Example 4: Two changes in a row

Question: The price of a scooter goes up by 20%. Later, the new price goes down by 10%. What is the overall percentage change?

1. Start with 100. Any starting value works, and with 100 the final answer reads straight off as a percentage.
2. Up 20%: 20% of 100 is 20, so the price becomes 120.
3. Down 10%: this 10% is of the price just before it, 120, not of the starting 100. 10% of 120 is 12, so the price becomes 120 − 12 = 108.
4. From 100 to 108 is a rise of 8. The overall change is an increase of 8%, not 20 − 10 = 10%.
5. With multipliers it is one line: 1.2 × 0.9 = 1.08, which is 8% more than the start.

A shortcut for two changes in a row. If the first change is a% and the second is b%, the overall change is a + b + (a × b)/100 per cent.

• Write an increase as a positive number and a decrease as a negative number. Here a = +20 and b = −10.
• (a × b)/100 = (20 × −10)/100 = −200/100 = −2.
• Overall = 20 + (−10) + (−2) = +8. A positive answer is an overall increase; a negative answer is an overall decrease.
• The last part, (a × b)/100, is the change on the change: the 10% fall also takes 10% off the 20 that was added, which is 2.
• The order does not matter: down 10% and then up 20% also ends at 108.

Your turn: a price goes up 30% and then goes down 30%. Use the shortcut. Is the overall change zero?

Check: 30 + (−30) + (30 × −30)/100 = 0 − 9 = −9. The price ends 9% lower than it started, not unchanged. With multipliers: 1.3 × 0.7 = 0.91.

The same change, year after year. A town has 25,000 people, and its population grows by 4% each year. "Each year" means each 4% is worked out on the population at the start of that year, so these are changes in a row too.

1. After 1 year: 25,000 × 1.04 = 26,000.
2. After 2 years: 26,000 × 1.04 = 27,040. The second 4% is of 26,000, which is 1,040, not 1,000.
3. Your turn: use the shortcut to find the overall percentage rise over the two years. Check that it gives the same population.

Check: 4 + 4 + (4 × 4)/100 = 8.16%. Then 8.16% of 25,000 = 2,040, and 25,000 + 2,040 = 27,040.

A value that falls by r% each year, such as the value of a machine, works the same way: multiply by (100 − r)/100 once for each year.

The principle: each change in a row is a percentage of the value just before it, not of the starting value. So never just add the percentages. Multiply the multipliers, or use a + b + (a × b)/100 with a minus sign for each decrease.

Example 5: Working backwards to the original value

Question: (a) After a 12% pay rise, Ramesh earns ₹33,600 a month. What was his monthly salary before the rise? (b) After spending 28% of his savings on a phone, Mohan has ₹3,600 left. How much did he have at first?

Part (a): after a rise

1. Call the old salary 100%. The 12% rise is 12% of the old salary, so the new salary is 100% + 12% = 112% of the old salary.
2. So 112% of the old salary = ₹33,600.
3. 1% of the old salary = 33,600 ÷ 112 = ₹300.
4. 100% = 300 × 100 = ₹30,000. Check: 12% of 30,000 is 3,600, and 30,000 + 3,600 = 33,600.
5. In one line: original = new value × 100 ÷ (100 + r) = 33,600 × 100 ÷ 112 = ₹30,000.

Do not take 12% of ₹33,600 (that is ₹4,032) and subtract it. That gives ₹29,568, which is wrong: the 12% was of the old salary, not the new one.

Part (b): after a fall

1. What is left is 100% − 28% = 72% of his savings.
2. So 72% of his savings = ₹3,600.
3. Your turn: find 100% of his savings.

Check: 1% = 3,600 ÷ 72 = ₹50, so 100% = ₹5,000. Check: 28% of 5,000 = 1,400, and 5,000 − 1,400 = 3,600.

The principle: the percentage in the question is of the original value. After a rise of r%, the new value is (100 + r)% of the original; after a fall of r%, it is (100 − r)%. So divide the new value by 100 + r (or by 100 − r after a fall) to get 1% of the original, then multiply by 100. For Ramesh: 33,600 ÷ 112 = 300, and 300 × 100 = ₹30,000. When working backwards, never take r% of the new value and subtract it.

Example 6: Pass marks

Question: (a) To pass an exam, a candidate needs 36% of the maximum marks. Arjun scored 207 marks and failed by 9 marks. What are the maximum marks? (b) In another exam, the pass mark is 40% of the maximum. Meena scored 52% and was 54 marks above the pass mark. What were the maximum marks in that exam?

1. Arjun failed by 9 marks, so the pass mark is 207 + 9 = 216 marks.
2. 36% of the maximum marks = 216, so 1% = 216 ÷ 36 = 6 marks.
3. 100% = 6 × 100 = 600 marks.
4. For Meena, the 54 marks are the gap between her score and the pass mark. That gap is 52% − 40% = 12% of the maximum marks.
5. Your turn: 12% of the maximum marks = 54. Find the maximum marks.

Check: 1% = 54 ÷ 12 = 4.5 marks, so the maximum is 4.5 × 100 = 450 marks. Check: 52% of 450 = 234, 40% of 450 = 180, and 234 − 180 = 54.

The principle: when two percentages are of the same whole, the gap between them is a percentage of that whole too. The number of marks in the gap is that percentage of the whole. Find 1% from it, then 100%.

The same idea works for votes. In an election between two candidates, if the winner gets 58% of the valid votes, the loser gets the other 42%. So the winning margin is 58% − 42% = 16% of the valid votes.

Revision: percentages worth knowing by heart

These pairs come up again and again. The full table, with decimals, is in the Fractions, Decimals and Percentages lesson.

Use them both ways:

• 12.5% of 720 is 1/8 of 720, which is 720 ÷ 8 = 90.
• A rise of 12.5% multiplies by 1 + 1/8 = 9/8, which is 1.125.
• Multiply a row to get its multiples: 3/8 is 3 lots of 1/8, so 3/8 = 3 × 12.5% = 37.5%.
• x% of y is the same as y% of x. So 16% of 25 = 25% of 16 = 4.

Revision: the formulas in one place

In this table r is a percentage written as a plain number: for a 12% rise, r = 12.

Common mistakes

• Using the wrong base. A fall from ₹9,000 to ₹8,000 is 1,000 ÷ 9,000, not 1,000 ÷ 8,000. The base is the value before the change.
• Adding changes in a row. Up 20% and then down 10% is up 8%, not up 10%. The second change works on the new value.
• Taking the percentage of the new value when working backwards. After a 12% rise to ₹33,600, the old salary is 33,600 × 100 ÷ 112, not 33,600 minus 12% of 33,600.
• Mixing up percentage points and per cent. If a district's literacy rate goes up from 64% to 72%, it has risen by 8 percentage points (72 − 64). As a percentage change it has risen by 8 ÷ 64 × 100 = 12.5% of its old value. Read which one the question asks for.
• Mixing units. To find 50 paise as a percentage of ₹2, change ₹2 to 200 paise first.

Next

Profit and loss, simple and compound interest, and data interpretation are all built on percentages. For quick revision, keep the Percentage formula cheat sheet handy. A good next topic is profit and loss: read the Profit and loss formula cheat sheet, then try the Profit and Loss quiz.

Ready to test this lesson? Take the Percentage quiz

How this was made: Drafted with AI assistance; every formula and worked figure recomputed independently; spot-checked by the editor on 23 October 2026.