Mental shortcuts
Tricks that make hard-looking products quick.
Best in your head · no pen, no calculator why?
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the idea
A few products look hard but follow a pattern that makes them quick. Each shortcut is ordinary algebra in disguise, so it always works when its pattern fits.
more of the idea
The neatest one squares a number ending in 5. Multiply the number in front of the 5 by one more than itself, then write 25 after it. For 65², 6 × 7 = 42, so the answer is 4,225. It works because 65 × 65 = 60 × 70 + 25.
techniques
Times 11: sum in the middle
- Write the two digits with a gap between them.
- Put their sum in the gap.
- If the sum is 10 or more, write its last digit and carry the 1 to the left digit.
- It works because × 11 is × 10 plus the number once more.
86 × 11
- 8 + 6 = 14, so there is a carry.
- Write the 4 in the middle and add 1 to the 8.
- That gives 9, 4, 6.
Answer: 946
Around a round number
- Find the round number halfway between them, and the distance to each.
- Square the round number.
- Take away the distance squared.
47 × 53
- Both are 3 away from 50.
- 50 × 50 = 2,500, and 3 × 3 = 9.
- 2,500 − 9 = 2,491.
Answer: 2,491
Near 100: cross over
- Find how far each number is from 100.
- Below 100, take one number's distance off the other number. Above 100, add it instead.
- That result is the number of hundreds.
- Add the two distances multiplied together.
96 × 93
- Short by 4 and 7.
- 96 − 7 = 89, so 8,900.
- 4 × 7 = 28.
- 8,900 + 28 = 8,928.
Answer: 8,928
watch out for
- Forgetting the carry in × 11. When the digits add to 10 or more, the 1 goes onto the left digit: 74 × 11 is 814.
- Squaring the front digit for a number ending in 5. For 65² it is 6 × 7, not 6 × 6.
- Adding the distance squared instead of taking it away. 48 × 52 is 4 less than 2,500, not 4 more.
- Dropping the last step near 100. For 97 × 96, the 3 × 4 = 12 still has to be added.
practice
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× 11
worked example
50 × 11
- Put the digit sum between the digits: 5 + 0 = 5.
- So 5, 5, 0: 550.
Answer: 550
Squares ending in 5
worked example
75²
- Take the tens part times one more: 7 × 8 = 56.
- Then write 25 after it: 5,625.
Answer: 5,625
Around a round number
worked example
23 × 17
- Both are 3 away from 20: 20² − 3² = 400 − 9 = 391.
Answer: 391
Near 100
worked example
99 × 95
- Both are near 100, short by 1 and 5.
- Cross-subtract: 99 − 5 = 94, and 94 hundreds is 9,400.
- Multiply the shortfalls: 1 × 5 = 5. 9,400 + 5 = 9,405.
Answer: 9,405