Percentages made clear.
A percentage means “out of 100”. Identify the base first, then convert the percentage into a multiplier. This also makes increases, discounts and reverse-percentage questions easier.
Key concepts in Percentages
Percentage of a value
x% of N = xN/100.
Percentage change
(new−old)/old × 100; the original value is the base.
Successive changes
Multiply factors: +20% then −10% means 1.2×0.9 = 1.08, an 8% increase.
Reverse percentage
If a final amount is 120% of the original, divide it by 1.2 to recover the original.
Useful fraction–percentage pairs
| Fraction | Percentage | Decimal |
|---|---|---|
| 1/2 | 50% | 0.5 |
| 1/4 | 25% | 0.25 |
| 1/5 | 20% | 0.2 |
| 1/8 | 12.5% | 0.125 |
| 3/4 | 75% | 0.75 |
A reliable solving method
- Name the quantity that represents 100%.
- Convert the rate into a decimal or useful fraction.
- Apply each change to its correct base.
- Estimate whether the answer should be larger or smaller than the starting value.
Step-by-step solved examples
Find 15% of 240.
- 10% is 24 and 5% is 12.
- Add them: 24+12 = 36.
- Equivalently, 240×0.15 = 36.
A price rises from ₹800 to ₹920. Find the percentage increase.
- Increase = 920−800 = ₹120.
- Use the original ₹800 as the denominator.
- 120/800×100 = 15%.
After a 20% discount, a product costs ₹640. Find marked price.
- ₹640 is 80% of marked price.
- Original = 640/0.8.
- Marked price = ₹800; the discount is ₹160.
Percentages: practice questions
Choose one answer before opening the explanation. Write down your reasoning as well as the option; a correct guess is different from a method you can repeat.
25% of 360?
Show answer and explanation — question 1
Answer C: 90
25% = 1/4, and 360/4 = 90.
50 rises to 60. Increase?
Show answer and explanation — question 2
Answer C: 20%
10/50×100 = 20%.
200 falls by 15%. New value?
Show answer and explanation — question 3
Answer C: 170
200×0.85 = 170.
Increase 10% then decrease 10%: net result?
Show answer and explanation — question 4
Answer B: 1% decrease
1.1×0.9 = 0.99.
30 is what percent of 120?
Show answer and explanation — question 5
Answer B: 25%
30/120×100 = 25%.
After a 25% increase, value is 500. Original?
Show answer and explanation — question 6
Answer B: 400
500/1.25 = 400.
Frequently asked questions
Can a percentage exceed 100?
Yes. 150% of a quantity is one and a half times it.
Why do equal increases and decreases not cancel?
They apply to different bases. The second change acts on the already changed amount.
Name the quantity that represents 100%. Estimate whether the answer should be larger or smaller than the starting value.