Multiplication · Technique guide
Mental Multiplication of Large Numbers
By Robert Hansson, Kung Fu Math
Multiplication tables up to 10 × 10 are the alphabet of mental arithmetic. Larger products are words built from that alphabet. You rarely need to memorise them; you need to know how to break them apart and put them back together.
The techniques below range from the universal, such as splitting a number into tens and ones, to special shortcuts that only work for certain numbers. Learn the universal method first, then add the shortcuts one at a time.
01
Split into parts
Break one factor into tens and ones, multiply each part and add the results. This is the distributive law, and it works for every multiplication. Choose to split the number that gives the easiest parts.
Worked example
47 × 6
- → 40 × 6 = 240
- → 7 × 6 = 42
- → 240 + 42 = 282
Answer: 282
Practice: Try 38 × 7, 64 × 9 and 125 × 8.
02
Multiply by 11
For a two-digit number, add its two digits and put the sum in the middle. If the sum is 10 or more, carry 1 to the first digit.
Worked example
57 × 11
- → 5 + 7 = 12
- → Put 2 in the middle and carry 1: 5 + 1 = 6
- → 627
Answer: 627
Practice: Try 36 × 11, 82 × 11 and 99 × 11.
03
Multiply by 5 and 25
5 is 10 ÷ 2, so multiply by 10 and halve. 25 is 100 ÷ 4, so multiply by 100 and halve twice. These turn hard products into easy ones.
Worked example
84 × 25
- → 84 × 100 = 8 400
- → Half: 4 200
- → Half again: 2 100
Answer: 2 100
Practice: Try 46 × 5, 132 × 5 and 36 × 25.
04
Square numbers ending in 5
Take the tens digit, multiply it by the next number up, and write 25 after it. This works because (10a + 5)² = 100·a(a + 1) + 25.
Worked example
75²
- → 7 × 8 = 56
- → Write 25 after it: 5 625
Answer: 5 625
Practice: Try 35², 65² and 105².
05
Difference of squares
When two numbers sit equally far from a round number, use (a + b)(a − b) = a² − b². 48 × 52 is 50² minus 2².
Worked example
48 × 52
- → Both are 2 away from 50
- → 50² = 2 500
- → 2² = 4
- → 2 500 − 4 = 2 496
Answer: 2 496
Practice: Try 39 × 41, 97 × 103 and 28 × 32.
06
Two-digit by two-digit
Split one factor and multiply the other by each part. Keep a running total from the largest part. With practice, 34 × 27 takes only two steps.
Worked example
34 × 27
- → 34 × 20 = 680
- → 34 × 7 = 238
- → 680 + 238 = 918
Answer: 918
Practice: Try 23 × 46, 58 × 32 and 71 × 19 (use 71 × 20 − 71).
Common mistakes
- Forgetting to multiply both parts. In 47 × 6, both 40 and 7 must be multiplied by 6.
- Forgetting the carry in the 11 trick. 57 × 11 is 627, not 5 127.
- Using difference of squares when the numbers are not equally far from the middle.
Summary
- Split into tens and ones and add the partial products.
- Turn × 5 and × 25 into × 10 or × 100 followed by halving.
- Look for patterns: numbers ending in 5, numbers around a round middle, and × 11.
Into the dojo
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