The 5 Best Mental Math Tricks

Key takeaways

• The five tricks worth learning first: multiplying by 11, squaring any number that ends in 5, multiplying two numbers that straddle a round ten, multiplying two-digit numbers whose tens match and whose ones add to 10, and subtracting anything from 1,000.

• All five are surprising the first time and take about a minute to learn. None of them require you to be fast at arithmetic already.

• 85² = 7,225 in one step: multiply 8 by 9 to get 72, then write 25.

• They work by cutting working memory load. Adult working memory holds roughly four items at a time, and the school method of stacking partial products spends most of that budget on bookkeeping.

• Each trick has a precondition. Recognizing when it applies is the skill that takes practice, and it is the part a list of tricks cannot teach you.

• Numbio teaches 48 of these shortcuts and then drills each one on problems where it actually applies.

What Are the 5 Best Mental Math Tricks?

The five below are the ones that combine the biggest surprise with the smallest learning cost: multiply by 11, square anything ending in 5, use the midpoint when two numbers straddle a round ten, exploit matching tens with ones that add to 10, and subtract from 1,000 by going through 9. Four are multiplication, one is subtraction, and every one of them can be learned in about a minute.

Deliberately left out are the shortcuts that feel like tricks but are really just definitions. Multiplying by 8 by doubling three times works, and so does multiplying by 10 by adding a zero, but nobody is surprised by either. The five here change the shape of the problem.

Why do these tricks work at all?

They reduce how much you have to hold in your head at the same time. There is no hidden mathematics in any of them: each is the distributive property or the difference of squares wearing a disguise, trading a step you find hard for a step you find easy.

That matters because the bottleneck in mental arithmetic is storage rather than calculation. Nelson Cowan's review of the evidence put the central capacity limit at about four chunks, revising the older "seven plus or minus two" figure downward (Behavioral and Brain Sciences, 2001). The school method of multiplying, right to left with carries, spends that budget fast: you compute a digit, hold a carry, hold a partial product, and start again while all of it is still live.

Researchers have measured the difference directly. Comparing the right-to-left school algorithm with the left-to-right method used by fast mental calculators, Rosenberg-Lee, Lovett and Anderson found the two approaches require essentially the same calculations but place different demands on working memory, with the school method demanding more (Cognitive, Affective, & Behavioral Neuroscience, 2009). Same arithmetic, different load.

1. How do you multiply any two-digit number by 11?

Add the two digits and drop the sum in the middle. For 17 × 11: 1 + 7 = 8, so the answer is 1, then 8, then 7, giving 187.

When the two digits add to more than 9, carry the extra into the leading digit. For 78 × 11: 7 + 8 = 15, so you get 7, then 15, then 8, and the 1 carries left to make 858. Numbio splits these into separate lessons for exactly that reason, because the no-carry version up to 18 is the one that clicks instantly and the carry is a second thing to learn.

This is the trick most people show someone else within a day of learning it.

2. How do you square a number ending in 5?

Multiply the leading digits by the next number up, then write 25 on the end. For 85²: 8 × 9 = 72, then 25, giving 7,225. For 65²: 6 × 7 = 42, then 25, giving 4,225.

It works at any length, so 105² is 10 × 11 = 110, then 25, giving 11,025. Two seconds of work for a calculation that looks like it should take twenty, and there is no precondition to check beyond the number ending in 5.

3. How do you multiply two numbers that straddle a round ten?

When two numbers sit the same distance either side of a multiple of 10, square the midpoint and subtract the square of the gap. For 66 × 74, the midpoint is 70: 70² = 4,900, the gap is 4, so 4² = 16, and 4,900 − 16 = 4,884.

The same move handles 18 × 22 (400 − 4 = 396) and 47 × 53 (2,500 − 9 = 2,491). This is the difference of squares, and it turns a genuinely unpleasant problem into two easy ones. Numbio files it under "Ten in the middle", which is a good name for what you are looking for: a round number sitting exactly between your two factors.

4. How do you multiply numbers like 54 × 56 instantly?

When two two-digit numbers have the same tens digit and their ones digits add to 10, multiply the tens digit by one more than itself, then multiply the two ones digits and stick the result on the end as two digits. For 54 × 56: 5 × 6 = 30, and 4 × 6 = 24, so the answer is 3,024.

For 43 × 47: 4 × 5 = 20, and 3 × 7 = 21, giving 2,021. The padding matters when the second product is a single digit. For 91 × 99: 9 × 10 = 90, and 1 × 9 = 9, which has to be written as 09, giving 9,009. Numbio calls this one "Front equal, back ten", and it is the trick that most reliably makes people ask whether it really works for every pair. It does.

5. How do you subtract from 1,000 without borrowing?

Subtract every digit from 9, except the last one, which you subtract from 10. For 1,000 − 728: 9 − 7 = 2, 9 − 2 = 7, 10 − 8 = 2, giving 272.

Try 1,000 − 456: 9 − 4 = 5, 9 − 5 = 4, 10 − 6 = 4, giving 544. The one wrinkle is a subtrahend ending in zero, where you drop the zero from both numbers first, subtract, then put it back: 1,000 − 730 becomes 100 − 73 = 27, so 270. Useful far more often than it sounds, because change, tips and rough budgeting are full of round thousands.

Where Numbio fits

Most people meet tricks like these as a list, memorize two, and lose them within a week. A list teaches the method but never the recognition, and recognition is the slow part. Executing 66 × 74 as 4,900 − 16 takes a second. Noticing that 66 and 74 straddle 70 is what needs repetition.

That gap is what Numbio is built for. It teaches 48 shortcuts explicitly, then keeps serving problems where each one applies, so you drill the spotting and not just the method. Every trick above is in there with a worked example and its own exercises, difficulty adapts as you go, and it's free on iPhone and iPad with the core exercises open to everyone. The features page has the full breakdown.

The bottom line

Start with multiplying by 11 and squaring numbers ending in 5. Both apply immediately, neither has a precondition worth thinking about, and both are surprising enough that you will use them. The other three pay off once you can spot their patterns without looking, which takes a few weeks of short sessions rather than a single sitting.

None of this replaces knowing your times tables. Every trick above ends in a single-digit fact, and if that fact is slow to arrive, the shortcut adds a step instead of removing one. If you are still weighing whether mental arithmetic is worth the effort at all now that every phone has a calculator, we made that case separately in Why Learn Mental Math When Calculators Exist?