Mental Maths - Calculate Faster in Your Head
Mental maths is the ability to perform calculations in your head – quickly, accurately, and without a calculator. It is one of the most practical skills in mathematics. From splitting a restaurant bill to estimating whether a budget stretches far enough, mental maths is used every single day. More importantly, building it strengthens number sense, pattern recognition, and mathematical confidence.
One of the most remarkable systems of rapid mental calculation was developed under extraordinary circumstances: the Russian engineer Jakow Trachtenberg devised his method of speed mathematics in the 1940s while imprisoned in a Nazi concentration camp, using it to keep his mind occupied and, he later said, to stay sane. After the war he founded the Mathematical Institute in Zurich to teach the system. Human mental-calculation ability can be astonishing: the Indian calculating prodigy Shakuntala Devi was recorded in the 1982 Guinness Book of World Records multiplying two 13-digit numbers entirely in her head in 28 seconds, faster than a contemporary computer of the era could confirm the answer.
Why Mental Maths Matters
- Saves time in everyday situations where reaching for a calculator is impractical.
- Builds a deep understanding of how numbers relate to each other.
- Develops the ability to estimate and spot errors in calculator output.
- Is directly tested in maths competitions, entrance exams, and job assessments.
Strategy 1: Partitioning
Break a number into its place-value parts, operate on each part, then recombine.
Example: 47 + 36 → (40 + 30) + (7 + 6) = 70 + 13 = 83.
Example: 67 × 4 → (60 × 4) + (7 × 4) = 240 + 28 = 268.
Strategy 2: Compensation
Round one number to a nearby convenient value, operate, then correct.
Example: 58 + 29 → 58 + 30 − 1 = 88 − 1 = 87.
Example: 83 − 38 → 83 − 40 + 2 = 43 + 2 = 45.
Example: 47 × 9 → 47 × 10 − 47 = 470 − 47 = 423.
Strategy 3: Doubling and Halving
Multiplying is easier when one factor is a power of 2. Double one number and halve the other to keep the product the same.
Example: 35 × 16 → 70 × 8 → 140 × 4 → 280 × 2 = 560.
Example: 25 × 44 → 50 × 22 → 100 × 11 = 1 100.
Works because (a × 2) × (b ÷ 2) = a × b.
Strategy 4: Number Bonds to 10, 100, and 1 000
Knowing pairs that sum to 10 (e.g. 3+7, 6+4) makes larger calculations automatic.
Example: 364 + 236 → 300+200=500, 64+36=100, total = 600.
Example: 1 000 − 387 → think: 387 + ? = 1 000. 387 → 400 (add 13), → 1 000 (add 600). Answer: 613.
Strategy 5: Left-to-Right Addition
Work from the largest place value to the smallest, building the answer progressively.
Example: 486 + 357:
400 + 300 = 700.
700 + 80 = 780.
780 + 50 = 830.
830 + 6 = 836.
836 + 7 = 843.
Strategy 6: Factors and Multiples in Multiplication
Replace a difficult multiplier with its factors, applied one at a time.
Example: 28 × 15 → 28 × 5 × 3 → 140 × 3 = 420.
Example: 36 × 25 → 36 × 100 ÷ 4 = 3 600 ÷ 4 = 900.
(Multiplying by 25 = multiplying by 100 then dividing by 4.)
Strategy 7: Multiplying by 5, 50, and 500
× 5: Multiply by 10 then halve. 68 × 5 = 680 ÷ 2 = 340.
× 50: Multiply by 100 then halve. 46 × 50 = 4 600 ÷ 2 = 2 300.
× 500: Multiply by 1 000 then halve. 34 × 500 = 34 000 ÷ 2 = 17 000.
Mental Division Strategies
| Divisor | Strategy | Example |
|---|---|---|
| ÷ 5 | Multiply by 2, divide by 10 | 840 ÷ 5 = 1 680 ÷ 10 = 168 |
| ÷ 25 | Multiply by 4, divide by 100 | 775 ÷ 25 = 3 100 ÷ 100 = 31 |
| ÷ 50 | Multiply by 2, divide by 100 | 3 500 ÷ 50 = 7 000 ÷ 100 = 70 |
| ÷ 4 | Halve, then halve again | 948 ÷ 4 = 474 ÷ 2 = 237 |
| ÷ 8 | Halve three times | 648 ÷ 8 = 324 ÷ 4 = 162 ÷ 2 = 81 |
Estimation
Round each number to its leading digit, compute, and adjust. Use estimation to check that a calculated answer is in the right ballpark.
Example: Is 47 × 83 roughly 3 900? → 50 × 80 = 4 000. Yes, reasonable.
Example: 389 ÷ 41 → ≈ 400 ÷ 40 = 10. Actual answer will be close to 10.
Key Takeaways
- Partitioning: split numbers by place value, operate, recombine.
- Compensation: round to a convenient number, operate, correct the adjustment.
- Doubling and halving keeps products the same while simplifying the calculation.
- ×5 = ×10 ÷ 2. ×25 = ×100 ÷ 4. ÷5 = ×2 ÷ 10.