Speed Maths - Calculation Tricks That Actually Work
Speed maths is mental arithmetic taken to the next level. Using clever algebraic identities and patterns, you can perform calculations that seem impossibly fast to observers – squaring three-digit numbers, multiplying near 100, and more – all in seconds. Every technique here has a solid mathematical reason behind it.
Technique 1: Squaring Numbers Ending in 5
For any number ending in 5: multiply the tens digit by (tens digit + 1), then append 25.
Formula: (10a + 5)² = a(a+1) × 100 + 25
35²: tens digit = 3. 3 × 4 = 12. Append 25 → 1 225.
65²: tens digit = 6. 6 × 7 = 42. Append 25 → 4 225.
95²: tens digit = 9. 9 × 10 = 90. Append 25 → 9 025.
Technique 2: Multiplying by 11
For a two-digit number ab: insert the sum of a and b between them.
If the sum exceeds 9, carry 1 into the hundreds digit.
Formula: 11 × (10a + b) = 100a + 10(a+b) + b
11 × 34: digits 3 and 4, sum = 7. Answer: 374.
11 × 72: digits 7 and 2, sum = 9. Answer: 792.
11 × 87: digits 8 and 7, sum = 15. Write 5, carry 1 to 8 → 9. Answer: 957.
Technique 3: Multiplying Two Numbers Near 100
Find how far each number is from 100 (the deficit). Cross-subtract one deficit from the other, then multiply the deficits for the last two digits.
Formula: (100 − a)(100 − b) = 100(100 − a − b) + ab
97 × 94: deficits are 3 and 6. Cross: 97 − 6 = 91 (or 94 − 3 = 91). Product: 3 × 6 = 18. Answer: 9 118.
96 × 93: deficits 4 and 7. Cross: 96 − 7 = 89. Product: 4 × 7 = 28. Answer: 8 928.
Technique 4: Squaring Numbers Near 50
For a number n = 50 + d (or 50 − d):
n² = 2 500 + 100d + d² (when n = 50 + d)
n² = 2 500 − 100d + d² (when n = 50 − d)
53²: d = 3. 2 500 + 300 + 9 = 2 809.
47²: d = 3. 2 500 − 300 + 9 = 2 209.
58²: d = 8. 2 500 + 800 + 64 = 3 364.
Technique 5: Difference of Two Squares
a² − b² = (a + b)(a − b)
This also reverses to simplify products: write any two numbers as (m+d) and (m−d), then their product = m² − d².
Example: 43 × 37 = (40+3)(40−3) = 40² − 3² = 1 600 − 9 = 1 591.
Example: 62 × 58 = (60+2)(60−2) = 3 600 − 4 = 3 596.
Technique 6: Multiplying Any Two-Digit Numbers (FOIL)
For (10a + b)(10c + d): compute ac, (ad + bc), bd. Combine with correct place values.
Example: 34 × 27:
3 × 2 = 6 (hundreds). 3 × 7 + 4 × 2 = 21 + 8 = 29 (tens). 4 × 7 = 28 (units).
= 600 + 290 + 28 = 918.
Technique 7: Multiplying Numbers Above 100
For numbers just above 100, use surpluses instead of deficits.
Example: 103 × 106: surpluses 3 and 6. Cross: 103 + 6 = 109. Product: 3 × 6 = 18. Answer: 10 918.
Example: 104 × 112: surpluses 4 and 12. Cross: 104 + 12 = 116. Product: 4 × 12 = 48. Answer: 11 648.
Why These Tricks Work
| Trick | Algebraic Identity Behind It |
|---|---|
| Square ending in 5 | (10a+5)² = 100a(a+1) + 25 |
| Multiply by 11 | 11(10a+b) = 100a + 10(a+b) + b |
| Near 100 multiplication | (100−a)(100−b) = 100(100−a−b) + ab |
| Difference of squares | (m+d)(m−d) = m² − d² |
| Near 50 squaring | (50±d)² = 2500 ± 100d + d² |
Key Takeaways
- Numbers ending in 5: square the tens digit × (tens+1), append 25.
- Multiply by 11: insert the digit sum between the two digits.
- Near 100: cross-subtract deficits for the hundreds, multiply deficits for the units.
- Difference of squares: write numbers as (m+d)(m−d) = m²−d².
Speed Drills
Aim to answer each in under 5 seconds:
- 45²
- 11 × 63
- 98 × 95
- 56² (near 50 method)
- 52 × 48 (difference of squares)
- 11 × 96
- 75²
- 103 × 107