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.
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.
Practice Drills
Try to answer each in under 10 seconds:
- 67 + 48 (use compensation)
- 125 × 8 (use doubling)
- 96 × 25 (use the ×100 ÷4 trick)
- 1 000 − 463
- 756 ÷ 4 (halve twice)
- 37 × 9 (use compensation: ×10 − 37)
- 54 × 15 (use factors: ×5 ×3)
- 2 800 ÷ 50