Mental Math Tricks That Actually Work, and the Conditions Where They Break

Most collections of mental arithmetic tricks share two problems: the examples are chosen to make the method look better than it is, and nobody tells you when it stops working. Every method below is worked through with arithmetic you can check line by line, and every one is stated with the case where it fails or becomes slower than the ordinary approach.

What separates a method from a party trick

A mental arithmetic shortcut is ordinary algebra with the middle steps precomputed. That is the whole story, and it has a practical consequence: every genuine method has a condition attached — a range of inputs where it applies and outside which it either stops helping or produces a wrong answer.

The tricks that circulate without their conditions are the ones that get people into trouble. So every method below is stated with the case where it fails, and every example has been worked so you can check it rather than take it on trust.

This article assumes you already have the foundations. Round-and-compensate, complements, the easy multipliers, multiplying by 11, squaring numbers ending in 5, difference of squares, one-percent anchoring and casting out nines are covered in our guide to mental math training. What follows is the next layer.

Multiplying numbers near 100

This is the highest-value method most people have never seen. It works when both numbers sit near a round base — 100 is the useful one for everyday arithmetic.

Both numbers below 100. Write down how far each falls short.

97 x 94 — shortfalls of 3 and 6.

1. Cross-subtract: 97 - 6 = 91. (Taking 94 - 3 gives the same 91, which is a built-in check.)
2. Multiply the shortfalls: 3 x 6 = 18.
3. Join them: 9,118.

Verify it the long way if you like: 97 x 90 = 8,730, 97 x 4 = 388, total 9,118.

Squaring is the same move. 97² — shortfall 3. So 97 - 3 = 94, and 3² = 9, giving 9,409.

Both numbers above 100. Use the excess and add instead of subtract.

108 x 106 — excesses 8 and 6. 108 + 6 = 114, and 8 x 6 = 48, giving 11,448. Check: 10,800 + 648 = 11,448.

The condition, and the case everyone gets wrong. The second part occupies two digit positions. If the product of the shortfalls reaches 100, it overflows and has to carry.

88 x 85 — shortfalls 12 and 15. 88 - 15 = 73. 12 x 15 = 180. Since 180 exceeds 99, the extra hundred carries: 73 + 1 = 74, leaving 80. Answer 7,480. Check: 88 x 80 = 7,040, plus 88 x 5 = 440, total 7,480.

That carry is why the method loses its appeal once the numbers drift far from 100. Within roughly ten either side it is very fast; beyond about fifteen you are doing a two-digit multiplication anyway.

Why it works. Expanding (100 - a)(100 - b) gives 10,000 - 100a - 100b + ab, which is 100(100 - a - b) + ab. The first bracket is the cross-subtraction and ab is the shortfall product — the algebra is just the binomial expansion with the arithmetic done in advance.

Same tens digit, units adding to ten

A close relative, and a generalisation of the squaring-numbers-ending-in-5 rule.

When two two-digit numbers share a tens digit and their units digits sum to 10, multiply the tens digit by the next whole number up, then append the product of the units digits.

43 x 47 — 4 x 5 = 20, 3 x 7 = 21 → 2,021. Check: 1,720 + 301 = 2,021.

62 x 68 — 6 x 7 = 42, 2 x 8 = 16 → 4,216. Check: 4,340 - 124 = 4,216.

Condition: both parts are required. Same tens digit and units summing to exactly 10. And the units product needs two digit places, so 61 x 69 gives 42 then 09 → 4,209, not 429. Squaring a number ending in 5 is simply the case where both units digits are 5.

Squaring an awkward number by shifting to an easy product

The difference-of-squares identity in the training guide turns a product into a square. Run it the other way and it turns an awkward square into an easy product plus a small correction: n² = (n - d)(n + d) + d².

Pick d to land on a round number.

43² → shift down 3 to 40: 40 x 46 = 1,840, plus 3² = 9 → 1,849. 68² → shift up 2 to 70: 70 x 66 = 4,620, plus 2² = 4 → 4,624.

Condition: you must add the correction when squaring, and the shift is worth it only if it produces a genuinely easy multiplier. If the nearest round number is more than about 5 away, the correction stops being a small number you can hold.

Division by awkward-looking divisors

Multiplication shortcuts get all the attention; the division versions are used less and are just as fast.

Each rests on the same observation: these divisors are factors of powers of ten, so replacing the division with a multiplication and a decimal shift is exact, not approximate.

Percentages that are really fractions

One-percent anchoring handles anything. But a handful of percentages are exactly simple fractions, and recognising them converts a multiplication into a division you already know.

Condition: the fraction route pays off only when the base divides cleanly. 37.5% of 97 is not made easier by eighths; fall back to one-percent anchoring there.

Percentage changes multiply — this is the important one

More everyday arithmetic goes wrong here than anywhere else in this article.

Sequential changes do not add. Start at 100. A 20% fall gives 80. A 20% rise on 80 gives 96, not 100. You are 4% down, because the rise was calculated on a smaller base than the fall.

Successive discounts do not add either. "30% off, then a further 20% off" is 0.7 x 0.8 = 0.56 — a 44% total reduction, not 50%.

Reversing a change means dividing. If a figure is 130 after a 30% increase, the original is 130 ÷ 1.3 = 100. If a price is 60 after 25% off, the original is 60 ÷ 0.75 = 80. Adding 25% to 60 would give 75, and 75 is not the answer.

The mental move is to stop thinking in additions and start thinking in multipliers: down 20% is ×0.8, up 30% is ×1.3, and going backwards means dividing by the multiplier you went forward with.