Mental Mathematics — Speed Calculation
Techniques for fast mental arithmetic — multiplication shortcuts, squaring, estimation, and the underlying patterns that make them work.
Why Mental Math
Calculators handle precision. Mental math handles speed — quick estimates, order-of-magnitude checks, spotting when something is obviously wrong. The goal isn’t to replace tools, it’s to stay ahead of them: formulate the right question before reaching for a calculator.
Multiplication Tricks
Multiplying by 9
Instead of multiplying by 9, multiply by 10 and subtract the number.
47 × 9 = 47 × 10 − 47 = 470 − 47 = 423
Multiplying by 11
For two-digit numbers: add the two digits and insert the sum in the middle.
35 × 11 → 3 _ 5 → 3+5=8 → 385
76 × 11 → 7 _ 6 → 7+6=13 → carry: 8, 3, 6 → 836
Multiplying near 100
For numbers close to 100, use the deviation from 100:
96 × 97
→ deviations: −4, −3
→ cross-subtract: 96−3 = 93 (or 97−4 = 93)
→ multiply deviations: 4 × 3 = 12
→ answer: 9312
Works because (100−a)(100−b) = 100(100−a−b) + ab.
Multiplying two-digit numbers
Split and distribute. 43 × 27:
= 43 × 20 + 43 × 7
= 860 + 301
= 1161
Or use the cross-multiplication method (Vedic: Urdhva-Tiryak):
4 3
× 2 7
──────
Step 1 (units): 3×7 = 21 → write 1, carry 2
Step 2 (cross): 4×7 + 3×2 = 28+6 = 34, +2 carry = 36 → write 6, carry 3
Step 3 (tens): 4×2 = 8, +3 carry = 11 → write 11
Result: 1161
Squaring
Numbers ending in 5
Square the first part n, then append 25.
75² → 7×8 = 56 → 5625
85² → 8×9 = 72 → 7225
Squaring near 50
Use (50+d)² = 2500 + 100d + d²
53² = 2500 + 300 + 9 = 2809
47² = 2500 − 300 + 9 = 2209
General squaring with the identity (a+b)²
Pick the nearest round number, find the deviation:
67² = (70−3)² = 4900 − 420 + 9 = 4489
Or use the difference of squares identity: a² = (a+d)(a−d) + d²
43² → nearest round: 40, d=3 → 46×40 + 9 = 1840 + 9 = 1849
Division and Fractions
Dividing by 5
Multiply by 2, then divide by 10 (i.e., shift decimal).
840 ÷ 5 = 1680 ÷ 10 = 168
Converting fractions to decimals
Memorise a small table and derive the rest:
| Fraction | Decimal |
|---|---|
| 1/8 | 0.125 |
| 1/6 | 0.1667 |
| 1/7 | 0.1428… |
| 1/9 | 0.111… |
| 3/8 | 0.375 |
3/7 = 3 × (1/7) = 3 × 0.1428 = 0.4285. Build from the unit fractions.
Estimation
Order of magnitude first
Before any calculation, nail the magnitude. 320 × 47: roughly 300 × 50 = 15,000. If your answer is 1,500 or 150,000, something’s wrong — catch it before going further.
Percentage shortcuts
- 10% — shift decimal left
- 5% — half of 10%
- 15% — 10% + 5%
- 1% — shift decimal left twice
- Any % — decompose into 10s and 1s
37% of 240:
30% = 72
7% = 16.8
= 88.8
Fermi estimation
When exact answers don’t matter, round aggressively and keep track of the power of 10. The skill is knowing which approximations compound and which cancel.
The Underlying Pattern
Almost every mental math trick is the same move: rewrite the problem into one you already know how to solve fast. Near a round number? Use deviation. Multiplication too complex? Distribute over addition. Fraction unfamiliar? Derive from a memorised unit fraction.
The tricks aren’t arbitrary shortcuts — they’re algebraic identities made fast through practice. Understanding the identity means you can reconstruct the trick on the fly, or derive one for a case you haven’t seen before.