Mental Strategies

You are at the shop. Two things cost £47 and £29. Have you got enough for a £80 note to cover it? There is no time to find a pen, and reaching for a calculator feels a bit silly for a sum this size. Good news: you don't need either. With a handful of mental strategies you can work this out in your head, faster than you could tap it into a phone.

The secret behind every trick is the same one: a hard sum is just an easy sum in disguise. Break the numbers into friendlier pieces, or nudge an awkward number to a tidy one — do the easy version — then tidy up at the end. Let's collect the tricks one at a time.

Trick 1 — Partitioning: split into tens and ones

Partitioning splits each number into its tens and its ones, adds the tens together and the ones together, then combines the two parts. This leans on place value: a 34 is really 30 and 4 travelling together.

34 + 28 = (30 + 20) + (4 + 8) = 50 + 12 = 62

Notice the ones added up past ten (4 + 8 = 12). That is fine — the extra ten just joins the other tens when you combine. Here is the shop sum the same way:

47 + 38 = (40 + 30) + (7 + 8) = 70 + 15 = 85

Partitioning is your reliable everyday workhorse: it works for any two numbers, and you never have to hold more than one small piece in your head at a time.

Trick 2 — Near doubles: lean on a double you know

Your brain knows its doubles by heart — 6 + 6 = 12, 8 + 8 = 16. A near double is a sum that sits right next to one of these, so you use the double and then nudge by one or two:

6 + 7 = (6 + 6) + 1 = 12 + 1 = 13 8 + 9 = (8 + 8) + 1 = 16 + 1 = 17

The same idea scales up: 25 + 26 is just (25 + 25) + 1 = 51. Whenever two numbers are close neighbours, reach for a double.

Trick 3 — Compensation: round, then adjust

Some numbers are just awkward. 29, 99, 48 — they sit annoyingly close to a tidy round number. Compensation rounds the awkward number up to the tidy one, does the easy sum, then pays back the little bit you borrowed.

Worked example — addition. To work out 47 + 29, round 29 up to 30. But 30 is 1 too many, so add it and then take one back:

47 + 29 = (47 + 30) - 1 = 77 - 1 = 76

Worked example — subtraction. Compensation shines on subtractions like 83 - 49. Taking away 49 is fiddly; taking away a tidy 50 is easy. So take 50 away — but that removed 1 too much, so hand it back:

83 - 49 = (83 - 50) + 1 = 33 + 1 = 34

Look carefully at the two adjustments. When you added too much, you subtracted at the end. When you took away too much, you added at the end. That flip is exactly where people trip up — the next card is all about it.

Trick 4 — Multiplying by shifting digits

Multiplying by 10 doesn't need any working at all: every digit slides one place to the left and a 0 fills the gap. 36 \times 10 = 360. By 100, they slide two places: 36 \times 100 = 3600.

That single move powers a whole family of shortcuts:

Worked example. A cinema ticket is £99 for a group of 6 — how much altogether? Reach for 6 \times 100 = 600, then give back the 6 you overcounted: 600 - 6 = 594. So £594, worked out before anyone finds the calculator.

Which trick, when?

You don't march through all four every time — you pick the one that makes this sum easiest. A quick guide:

With practice you stop choosing consciously — the right trick just leaps out at you, the way you recognise a friend's face.

Counting up beats taking away

Subtraction has one more shortcut worth its own moment. Instead of taking away, count up from the smaller number to the bigger one — the difference is simply how far you travelled. To work out 71 - 68, hop up from 68 to 71: that is 3 steps, so the answer is 3.

71 - 68 = 3 \quad (68 \to 69 \to 70 \to 71)

This is exactly how a shopkeeper gives change: they don't compute 50 - 37, they count coins up from 37 until they reach 50.

Five tricks for calculating in your head:

Compensation is the trick people get backwards, and it always happens at the very last step. If you rounded 29 up to 30 to add it, you have added 1 too much — so you must subtract 1 at the end, not add it:

47 + 29 = (47 + 30) - 1 = 76 \quad(\text{not } 78)

Subtraction flips the direction. To do 83 - 49 you take away 50 — but that removed 1 too much, so you add 1 back:

83 - 49 = (83 - 50) + 1 = 34 \quad(\text{not } 32)

The safety check that never fails: after rounding, ask yourself "did I add too much, or too little?" — and undo exactly that. Say it out loud every time until it becomes a habit.

At the Mental Calculation World Cup, competitors add long columns of eight-digit numbers, and multiply huge numbers together, faster than a judge can type them into a calculator. They are not born with magic brains — they use exactly the tricks on this page, just scaled up: partitioning enormous numbers into place-value chunks, and compensating around round thousands and millions.

The everyday payoff is smaller but real. Once "×5 is ×10 then halve" and "×9 on your fingers" live in your head, splitting a bill between friends, checking your change, and estimating a trolley of shopping stop being chores you reach for a phone to do — they become things you just know, on the spot. That is a genuinely useful superpower for the price of a little practice.