In this guide
"Per cent" means "out of 100". So 20% means 20 out of 100, or 20/100, or 0.2. Every percentage question is built on that one idea.
Why spend extra time here? Because percentage is not one topic. Profit and loss, discount, simple and compound interest, and many data questions are percentage questions in disguise. A candidate who is quick with percentages finds those chapters easy. One who is slow finds all of them hard.
Fractions you should know as percentages
| Fraction | % | Fraction | % |
|---|---|---|---|
| 1/2 | 50% | 1/8 | 12.5% |
| 1/3 | 33⅓% | 1/10 | 10% |
| 1/4 | 25% | 1/12 | 8⅓% |
| 1/5 | 20% | 1/20 | 5% |
| 1/6 | 16⅔% | 3/4 | 75% |
These let you turn a percentage into a quick division. 12.5% of 640 is simply 640 ÷ 8 = 80.
Finding x% of a number
x% of N = N × x ÷ 100
The faster way is to build from 10% and 1%:
- 10%: move the decimal point one place left. 10% of 450 = 45.
- 5%: half of 10%. 5% of 450 = 22.5.
- 1%: move the decimal point two places. 1% of 450 = 4.5.
- Build up: 35% = 10% + 10% + 10% + 5%.
What percentage is one number of another?
Percentage = (part ÷ whole) × 100
45 is what percentage of 180? 45/180 = 1/4 = 25%.
Increase and decrease
Percentage change = (change ÷ original value) × 100
Always divide by the original value, the one you started from.
To increase a value by r%, multiply by (100 + r)/100. To decrease by r%, multiply by (100 − r)/100. So a 20% rise is × 1.2 and a 15% fall is × 0.85.
Finding the original. If the new value is known, divide by the multiplier. After a 25% rise the value is 1.25 times the original, so original = new ÷ 1.25.
Two changes in a row
When one percentage change follows another, the second acts on the new value. For successive changes of a% and b%:
Net change = a + b + (a × b ÷ 100)
Use a minus sign for a decrease. A 20% rise followed by a 20% fall gives 20 − 20 + (20 × −20 ÷ 100) = −4%, a net fall of 4%, not zero.
| Situation | Net effect |
|---|---|
| +10% then +10% | +21% |
| +20% then −20% | −4% |
| −10% then −10% | −19% |
| +50% then −50% | −25% |
"More than" and "less than"
If A is 60 and B is 50:
- A is more than B by 10/50 × 100 = 20% (divide by B).
- B is less than A by 10/60 × 100 = 16⅔% (divide by A).
The two answers are different because the base is different. General rule: if A is r% more than B, then B is less than A by r ÷ (100 + r) × 100 per cent.
Solved examples
Example 1. Find 35% of 480.
- 10% = 48, so 30% = 144.
- 5% = 24.
- 35% = 144 + 24 = 168.
Example 2. After a 20% increase, a salary is ₹18,000. What was it before?
- New salary = 1.2 × old salary.
- Old = 18,000 ÷ 1.2 = ₹15,000.
- Check: 20% of 15,000 = 3,000, and 15,000 + 3,000 = 18,000.
Example 3. The price of an item rises by 20% and then falls by 20%. Find the net change.
- Take a base of ₹1,000.
- After +20%: ₹1,200. After −20% of 1,200 = 240: ₹960.
- Net change = 40 on 1,000 = 4% decrease.
Example 4. A candidate needs 40% to pass. They score 160 marks and fail by 40 marks. Find the maximum marks.
- Pass mark = 160 + 40 = 200.
- 40% of maximum = 200.
- Maximum = 200 × 100 ÷ 40 = 500.
Example 5. A town has 8,000 people. The population grows by 10% each year. Find it after 2 years.
- After year 1: 8,000 × 1.1 = 8,800.
- After year 2: 8,800 × 1.1 = 9,680.
- Note the second year's growth is on 8,800, not 8,000.
Example 6. The price of sugar rises by 25%. By what percentage must a family cut consumption to keep spending the same?
- Cut = r ÷ (100 + r) × 100 = 25 ÷ 125 × 100 = 20%.
- Check: price ₹40 becomes ₹50; 10 kg cost ₹400 before; ₹400 buys 8 kg now, a cut of 2 kg, which is 20%.
Common mistakes
| Mistake | Correct way |
|---|---|
| Dividing by the new value for % change | Divide by the original value |
| +20% then −20% = no change | Net is −4% |
| Original = new − r% of new | Original = new ÷ (1 + r/100) |
| A is 20% more than B, so B is 20% less than A | B is 16⅔% less |
| Adding yearly growth to the first year's base each time | Each year grows on the previous year's figure |
Practice set
- Find 15% of 640.
- 45 is what percentage of 300?
- A price rises from ₹250 to ₹300. Find the percentage increase.
- After a 10% decrease, a number becomes 270. Find the number.
- A salary rises by 10% and then by another 10%. Find the total percentage rise.
- A's income is 25% more than B's. By what percentage is B's income less than A's?
- The pass mark is 35%. A candidate scores 150 and fails by 25 marks. Find the maximum marks.
- A village of 20,000 people shrinks by 5% each year. Find its population after 2 years.
Answers:
- 10% = 64, 5% = 32, total 96.
- 45/300 × 100 = 15%.
- 50/250 × 100 = 20%.
- 270 ÷ 0.9 = 300.
- 10 + 10 + (10 × 10 ÷ 100) = 21%.
- 25 ÷ 125 × 100 = 20%.
- Pass mark = 175 = 35% of maximum, so maximum = 175 × 100 ÷ 35 = 500.
- 20,000 × 0.95 = 19,000; 19,000 × 0.95 = 18,050.
What to do next
- Learn the fraction–percentage table until it is automatic.
- Practise 20 "x% of N" questions using the 10% and 5% method, without writing.
- Apply percentage in profit and loss and discount.
- Revise fractions in our fractions and decimals guide if conversions feel slow.
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Staff Selection Commission website .
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