Percentages

Percentages appear in almost every aptitude test, and they are also the base for profit and loss, interest and data interpretation. The good news is that a handful of ideas cover most questions. Learn them once, and practise them until the arithmetic feels automatic.

What a percentage is

"Per cent" means "per hundred". So 35% means 35 out of 100, or 35/100 = 0.35. To find a percentage of a number, multiply the number by the percentage and divide by 100.

The three basic questions

1. Find x% of N: N × x / 100. Example: 18% of 350 = 350 × 18 / 100 = 63.
2. What percent is a of b: (a / b) × 100. Example: 72 is what percent of 90? 72/90 × 100 = 80%.
3. Find the number when x% of it is known: known value × 100 / x. Example: 30% of a number is 54, so the number is 54 × 100 / 30 = 180.

Fractions worth remembering

Turning a percentage into a fraction makes the sum quick. Learn these until you do not need to think:

• 50% = 1/2, 25% = 1/4, 75% = 3/4, 20% = 1/5, 10% = 1/10
• 33⅓% = 1/3, 66⅔% = 2/3, 16⅔% = 1/6, 8⅓% = 1/12
• 12.5% = 1/8, 37.5% = 3/8, 62.5% = 5/8, 6.25% = 1/16

For instance, 12.5% of 640 is one eighth of 640, which is 80, with no multiplication needed.

Percentage change

When a quantity goes from an old value to a new value:

Percentage change = (new − old) / old × 100.

The key rule: always divide by the old (original) value. A price that rises from 800 to 920 has changed by 120, and 120/800 × 100 = 15%, an increase of 15%.

Successive percentage changes

Two changes one after the other do not simply add. If a quantity changes by a% and then by b%, the net change is:

a + b + (a × b) / 100 (use negative numbers for decreases).

Example: an increase of 10% followed by an increase of 20% gives 10 + 20 + (10 × 20)/100 = 32%. An increase of 20% followed by a decrease of 20% gives 20 − 20 + (20 × −20)/100 = −4%, which is a 4% overall decrease. A rise and an equal fall never cancel out.

"More than" and "less than" are not symmetrical

If A is x% more than B, then B is less than A by x / (100 + x) × 100 %. If A is 25% more than B, B is 25/125 × 100 = 20% less than A. The base of the comparison changes, so the percentage changes too. The same idea gives the price and consumption rule: if a price rises by x%, consumption must fall by x/(100 + x) × 100 % to keep spending the same.

Worked examples

Example 1. In an exam with 600 total marks, Neha gets 462. What is her percentage? 462/600 × 100 = 77%.

Example 2. A shirt costs 1,250 after a 20% discount was given on the marked price. What was the marked price? The 1,250 is 80% of the marked price, so marked price = 1,250 × 100 / 80 = 1,562.50.

Example 3. A town's population of 50,000 grows 4% in the first year and 5% in the second. Net growth = 4 + 5 + 20/100 = 9.2%, so the population is 50,000 × 1.092 = 54,600.

Common mistakes

• Dividing by the new value instead of the old one when finding change.
• Adding successive percentages directly, for example treating +20% and −20% as zero.
• Mixing up "percent" and "percentage points". Going from 40% to 50% is a rise of 10 percentage points but a rise of 25 percent.
• Forgetting that a "pass mark" or "margin" is a difference of two percentages, which is itself a percentage of the total.

Exam tips

1. Convert to a fraction when the percentage is a friendly one (25%, 12.5%, 33⅓%).
2. For "less than the other" questions, ask yourself which value is the base.
3. Estimate first. If you are asked for 18% of 350 and your answer is 630, something has gone wrong, because 10% is only 35.

Try the quiz below. After each answer you will see the reason, so use the wrong answers to find which rule you need to revisit.