Mental Maths Techniques for Competitive Exams

The short answer: learn six or seven techniques well, drill them for ten minutes a day and apply them where a calculation would otherwise take a long time. The aim is not to be a human calculator but to avoid slow, error-prone arithmetic on the questions that matter.

Principles before techniques

• Break numbers into friendly parts. 47 × 12 is 47 × 10 plus 47 × 2.
• Use the structure of the question. A percentage of a number is often easier as a fraction of it.
• Estimate first. If options differ widely, a rough answer is enough.
• Check with a different route. Recompute a doubtful answer by a different method, not the same one.

Technique 1: Percentages as fractions

Memorise: 50% = 1/2, 25% = 1/4, 20% = 1/5, 12.5% = 1/8, 16.67% = 1/6, 14.29% = 1/7, 11.11% = 1/9, 10% = 1/10, 8.33% = 1/12, 6.25% = 1/16.

Example. What is 12.5% of 640? 640 ÷ 8 = 80.

Example. A price rises by 16.67% from 360. The increase is 360 ÷ 6 = 60, so the new price is 420.

Technique 2: Splitting multiplication

Example. 68 × 15 = 68 × 10 + 68 × 5 = 680 + 340 = 1,020.

Example. 49 × 22 = 50 × 22 − 22 = 1,100 − 22 = 1,078.

Example. 125 × 48 = (1,000 ÷ 8) × 48 = 1,000 × 6 = 6,000.

Technique 3: Squares

Numbers ending in 5. For n5², multiply n by (n + 1) and append 25. So 35² = 3 × 4 = 12, then 25: 1,225. And 85² = 8 × 9 = 72, then 25: 7,225.

Near a round number. 98² = (100 − 2)² = 10,000 − 400 + 4 = 9,604. 52² = (50 + 2)² = 2,500 + 200 + 4 = 2,704.

Memorise squares to 30 and cubes to 15: see squares, cubes and roots practice.

Technique 4: Divisibility and remainders

• By 3: the digit sum is divisible by 3. By 9: the digit sum is divisible by 9.
• By 4: the last two digits are divisible by 4. By 8: the last three digits.
• By 11: the difference between the sum of digits in odd and even places is 0 or a multiple of 11.

Example. Is 5,742 divisible by 3? 5 + 7 + 4 + 2 = 18, yes. By 9? 18 is divisible by 9, yes. By 11? (5 + 4) − (7 + 2) = 0, yes.

Technique 5: Successive percentage change

Use multiplying factors. An increase of 20% then a decrease of 10%: 1.2 × 0.9 = 1.08, so a net 8% increase. Two successive increases of 10% each: 1.1 × 1.1 = 1.21, so 21%.

Technique 6: Estimation and options

Example. What is 29.8% of 4,010? About 30% of 4,000 = 1,200. If the options are 1,195, 1,350, 1,500 and 1,650, pick 1,195 without exact calculation.

Estimation works when options are far apart. When two options are close, use a more precise method.

Technique 7: Averages by deviation

Example. The average of 48, 52, 47, 55 and 53. Take 50 as a base: deviations are −2, +2, −3, +5, +3, which sum to +5, so the average is 50 + 5 ÷ 5 = 51.

A ten-minute daily routine

Rotate through the techniques over the week, add one new technique every few days, and keep a card of the ones you forget. After a month, time yourself on a set of 20 mixed calculations and note the improvement.

Where they pay off

These techniques speed up percentage, profit and loss, simplification and data interpretation. See how to practise data interpretation faster for application in charts and tables.

Common mistakes

• Memorising tricks without understanding them, which fails when the numbers change.
• Using a trick where plain calculation is quicker.
• Practising untimed.
• Learning too many techniques at once.
• Skipping the check on the final answer.

To test the techniques in context, take a timed practice test on quantitative topics.