David Lemm

Mental arithmetic can be faster (not to mention more impressive) than using your phone – all it takes is a few tricks.

"Calculate left-to-right in your head, not right-to-left as you might on paper," says "mathemagician" Arthur Benjamin, co-author of Secrets Of Mental Math (Three Rivers Press). "After that, it just takes practice."


Cut the problem into small bits

Even the most daunting of mental-arithmetic problems can normally be separated into easy ones – all it takes is a little addition or subtraction. "For a very simple but practical example: it's easy to calculate a ten percent tip. How do you calculate a 15 per cent tip?

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Well, take the ten per cent tip, cut it in half and add it to the original. So if the bill is £40, ten per cent of that is £4, half again is an additional £2. So 15 per cent of £40 = £6." The same principle would apply to 17.5 per cent.

Pay attention to complements


A complement is the distance between a number and the nearest convenient round number – normally ending in zeroes. "So the complement of 78 is 22 – it's the number you add to reach 100," says Benjamin. "On paper, 1234 minus 678 is messy. But you start by over-subtracting: 1234 minus 700 equals 534. I've subtracted too much, so I have to add back the complement. How much was 700 above 678? 22. You've turned a hard subtraction problem into an easy addition problem: 534 plus 22 equals 556.

Conquer and divide

"If you have a seven-digit number divided by a three-digit number, the answer is going to be a four-digit number or possibly one more," says Benjamin. (The same trick can be used when multiplying – just add instead.) "Therefore, if you were going to calculate 4,000,000 divided by 600, your answer is going to have four digits. Then simplify it: four-sixths is like two-thirds, which is about 0.667. You know the length, therefore you know your answer is going to be in the vicinity of 7,000." (It's 6,666.667.)

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A tip to be square

Here's a trick to calculate square numbers quickly. What's 242? It's simpler than it looks at first glance: "Go down four to the nearest easy number: 20. To balance it, go up four to 28. So the first calculation is 20 times 28. Two times 28 = 56, so add the zero and the answer is 560. You're close. To get the final result, all you have to do is to add the square of the number you went up or down – in this case it was four, to get your answer: 576. So, to summarise, 24 x 24 = (20 x 28) + (4 x 4), which is 576."

Take shortcuts whenever possible


"There are certain types of problems for which there are shortcuts to get an answer quickly," says Benjamin. To multiply any number by 11, add the sum of its digits and then insert it between your original numbers. "To work out 53 times 11, you'd add five plus three, so the answer is 583." To square any two-digit number that ends in five, the answer will always end in 25. "You take the first digit and multiply that by its next higher digit.

For example, 352: three times four equals 12, and it ends in 25. So your answer is 1,225."

This article was originally published in April 2014.