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Maths: Calculating in Your Head

▶  Watch: how to calculate in your head5 min

Four steps in about five minutes — one real question, worked step by step, then the traps to watch out for.

Some sums are quicker to work out in your head than in columns. You get one sum, like 22 × 19, and five answers to choose from.

There's no calculator, and only one of the five answers is right.

In GL's papers these are bare sums or one-line stories: times or divide by 10, 100 or 1,000, a times-table fact inside a bigger number, a number just below a round one, or "how much more?". The video above and the worked example below work through the same question, with the options in the same order. Then both work a second, short question, where the fix goes the other way.

The method

Four steps: find a round number, do the easy sum, fix what you changed, check it another way.

  1. Step 1 of 4

    Find a round number

    Look for a number that's close to 10, 100 or 1,000.
  2. Step 2 of 4

    Do the easy sum

    Use the round number instead. Then ask: will the real answer be a bit more, or a bit less?
  3. Step 3 of 4

    Fix what you changed

    Ask which way, and by how much.
  4. Step 4 of 4

    Check it another way

    Split a number, or work it backwards.

Worked example

What is 22 × 19?

A. 418 B. 421 C. 428 D. 440 E. 462

  1. Find a round number. 19 is very close to 20. It's just 1 less.
  2. Do the easy sum. 22 × 2 = 44, so 22 × 20 = 440. 19 is less than 20, so the real answer must be a bit less than 440. D is 440 itself, so D is out. E is even bigger, so E is out too.
  3. Fix what you changed. 20 lots of 22 is one lot too many, so take one 22 away: 440 − 22 = 418. B took away 19, not 22, so B is out.
  4. Check it another way. Split 22 into 20 and 2. 20 × 19 = 380, and 2 × 19 = 38. Together that's 418 again. C is ten more than that, so C is out. So the answer is A.

Answer: A — 418 (22 × 20 = 440, then take one 22 away)

A second worked example

Sometimes the fix goes the other way.

What is 7,031 − 2,999?

A. 4,031 B. 4,032 C. 4,042 D. 5,968 E. 10,030

  1. Find a round number. 2,999 is 1 less than 3,000.
  2. Do the easy sum. 7,031 − 3,000 = 4,031, so the answer is about 4,000. E is bigger than the number we started with, but taking away makes it smaller, so E is out. D is nearly 6,000, far from 4,000, so D is out too.
  3. Fix what you changed. You took away 1 too many, so add 1 back: 4,032. A never added it back, so A is out.
  4. Check it another way. Add it back on: 4,032 + 2,999 = 7,031. C gives ten too many, so C is out. The answer is B.

Answer: B — 4,032 (7,031 − 3,000 = 4,031, then add 1 back)

Things to watch out for

  1. Count up across a round number. Two jumps, not one. Ana needs 5,009 points, and she has 4,994. How many more does she need? Count up from the smaller number: 6 takes you to 5,000, then 9 more takes you to 5,009. 6 + 9 = 15. D and E each count only one of the jumps.
  2. Count the zeros. A whole number × 1,000 gains three zeros. A coin is 4 mm thick. How tall is a pile of 1,000 coins? 1,000 has three zeros, so 4 becomes 4,000: 4,000 mm. A has four zeros, one too many, and D has only one.
  3. Take the same zeros off both. Then check by multiplying back. What is 4,200 ÷ 70? Take one zero off each number: 420 ÷ 7 = 60. Check it: 70 × 60 = 4,200. 6 is too small, because 70 sixes are only 420. And 600 is far too big.

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