Key takeaways
- Every worthwhile mental multiplication shortcut is an application of the distributive property, which is why they generalise rather than being isolated tricks.
- Doubling and halving preserves the product and turns awkward numbers into easy ones.
- Multiplying by 11 works because 11 = 10 + 1, so the shortcut is just addition of shifted digits.
- Squaring numbers that end in 5, and multiplying pairs that are equidistant from a round number, both come from the difference-of-squares identity.
- Knowing why a shortcut works means you can tell when it applies, which is the difference between a technique and a party trick.
Nearly every mental multiplication shortcut is the distributive property in disguise. Break one number into parts that are easier to handle, multiply each part, add the results. Once you see that, the tricks stop being a list to memorise and become one idea in several shapes.
The general method
a × (b + c) = (a × b) + (a × c)
That is it. Every technique below is a choice about how to split a number so that the pieces are easy.
Example. 7 × 48. Split 48 into 50 − 2. 7 × 50 = 350, 7 × 2 = 14, so 350 − 14 = 336.
Splitting toward a round number is almost always the right instinct.
Doubling and halving
Halve one number, double the other. The product is unchanged, because you multiplied by two and divided by two.
Example. 14 × 35. Halve 14, double 35: 7 × 70 = 490.
When it helps: when one number is even and halving it produces something you know your times table for, or when doubling the other produces a round number.
When it does not: when both numbers are odd. Halving produces a fraction and you have made it worse.
You can apply it repeatedly. 16 × 25 → 8 × 50 → 4 × 100 = 400.
Multiplying by 11
The familiar version for a two-digit number: add the digits and put the sum in the middle.
Example. 34 × 11. 3 + 4 = 7, so 374.
When the digits sum to more than nine, carry into the leading digit. 57 × 11: 5 + 7 = 12, so 5|12|7 becomes 627.
Why it works. 11 = 10 + 1, so 34 × 11 = 340 + 34 = 374. The digit-sum pattern is what that addition looks like when written out.
Knowing the reason means you can handle three-digit numbers too, where the memorised version breaks down: 234 × 11 = 2340 + 234 = 2574.
Squaring numbers ending in 5
Take the digits before the 5, multiply by one more than that number, append 25.
Example. 35². 3 × 4 = 12, so 1225. Example. 85². 8 × 9 = 72, so 7225.
Why it works. A number ending in 5 is 10n + 5. Expanding:
(10n + 5)² = 100n² + 100n + 25 = 100n(n + 1) + 25
The "100 × n(n+1)" is the first part shifted two places; the 25 is appended.
Numbers equidistant from a round number
Two numbers the same distance either side of a round number multiply neatly.
Example. 47 × 53. Both are 3 away from 50. 50² − 3² = 2500 − 9 = 2491.
Why it works. This is the difference of squares: (a − b)(a + b) = a² − b².
This one is genuinely useful because the pattern is common — 18 × 22, 96 × 104, 35 × 45 — and it converts a hard multiplication into a square you probably know minus a small square.
Near a round number, generally
When both numbers are close to 100, there is a fast method.
Example. 96 × 97. Deficits from 100: 4 and 3. First part: 96 − 3 = 93 (or equivalently 97 − 4). Second part: 4 × 3 = 12. Answer: 9312.
Why it works. (100 − a)(100 − b) = 10000 − 100(a + b) + ab. The first part is 100 − (a + b) in hundreds; the second is ab.
Watch the second part: if ab exceeds 99, carry into the first part.
What to actually learn
Not all of them. Two or three you use often beat a dozen you half-remember and misapply.
The ones we would keep:
- Splitting toward a round number. The general method; covers most cases.
- Doubling and halving. Cheap and frequently applicable.
- Difference of squares for equidistant pairs. Because the pattern is common and the alternative is genuinely hard.
The rest are worth understanding once, so you recognise them when they apply.
The reason understanding matters
A memorised trick works on the cases it was memorised for and fails silently outside them. Someone who knows the 11 shortcut as a digit pattern gets 57 × 11 wrong; someone who knows it as 10 + 1 does not.
Knowing why a shortcut works tells you when it applies. That is the difference between a technique and a party trick.
Practising this
These respond to short, frequent practice with immediate feedback — retrieval and spacing, both well supported in the learning literature (Dunlosky et al. on technique effectiveness, Cepeda et al. on distributed practice).
Mental Math & Memory Games is built for that format — short sessions, immediate right-or-wrong feedback, repeated over time. It is free to download on Google Play, supported by ads, with optional in-app purchases.
We are not claiming this makes you smarter generally. The evidence for transfer from arithmetic practice to unrelated cognitive abilities is weak, and working memory and training covers that literature including where it undercuts claims made in this app category. What practice reliably improves is arithmetic.
Mental math tricks that work covers the wider set, estimation techniques covers the approximate side, and percentages in your head covers the calculation people need most often. The rest is under education and brain games.
Good to know
Frequently asked questions
Why do multiplication shortcuts work?
What is doubling and halving?
Why does the multiply-by-11 trick work?
How do you square a number ending in 5?
Do I need to memorise all of these?
Sources
- Improving Students' Learning With Effective Learning Techniques (Dunlosky et al., Psychol Sci Public Interest 2013;14(1):4–58) (opens in a new tab)PubMed, U.S. National Library of Medicine — accessed
- Distributed practice in verbal recall tasks: A review and quantitative synthesis (Cepeda et al., Psychol Bull 2006;132(3):354–80) (opens in a new tab)PubMed, U.S. National Library of Medicine — accessed