Technique explanation
Multiplying or dividing by 5, 25, or 125 can be simplified by using their fractional relationships to powers of ten (aliquot parts). Since 5 = 10/2, 25 = 100/4, and 125 = 1000/8, we can replace difficult multiplication with easier division (and vice versa) combined with place-value shifts.
Method Condition: Best for numbers that are easily divisible by 2, 4, or 8.
Standard fallback: Standard long multiplication or division using conventional algorithms.
Worked examples
Example 1: Multiplication by 25
Calculate 736 × 25.
- Instead of × 25, divide by 4: 736 ÷ 4 = 184.
- Multiply by 100 (shift two places): 184 → 18400.
- Result: 18400.
- Check: 736 × 100 ÷ 4 = 73600 ÷ 4 = 18400.
Example 2: Division by 125
Calculate 8624 ÷ 125.
- Instead of ÷ 125, multiply by 8: 8624 × 8 = 68992.
- Divide by 1000 (shift three places): 68992 → 68.992.
- Result: 68.992.
- Check: 68.992 × 125 = 8624 (Inverse multiplication confirms the quotient).
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