Key takeaways
- Practice reliably makes you faster and more accurate at the thing you practise — that is a narrow claim, it is the honest one, and arithmetic fluency is worth having on its own terms.
- Almost every mental arithmetic method is one of three ideas: move to a round number and compensate, break a number apart and recombine, or replace an operation with an easier equivalent.
- Casting out nines catches most arithmetic slips in about two seconds, but it is blind to transposed digits and to errors that happen to be multiples of nine.
- Percentages get much easier once you know that x percent of y always equals y percent of x.
- Short daily sessions beat long weekly ones, because the thing being trained is recall speed and recall speed decays between sessions.
There is a version of this article that opens by telling you arithmetic drills will rebuild your brain. This is not that article, because that claim is not one anyone can currently make honestly.
Here is what practice does, stated narrowly enough to be true: it makes you faster and more accurate at the thing you practised. Drill multiplication and your multiplication gets quicker and your error rate falls. Whether that carries over to abilities you did not train is a genuinely contested research question, and any app promising you general intelligence, protection from decline, or transfer to unrelated skills is selling past the evidence.
Narrow is still worth having. Being able to split a bill, sanity-check a spreadsheet, or work out whether a discount is actually a discount without reaching for a phone is a real, daily, slightly embarrassing gap for a lot of adults. That is the thing this fixes.
Three ideas behind almost every method
Mental arithmetic looks like a bag of unrelated tricks. It is mostly three moves.
- Move to a round number, then compensate. Round numbers are easy; the correction is small.
- Break a number apart, then recombine. The distributive law, used deliberately.
- Swap the operation for an easier equivalent. Halve and double, multiply instead of divide, use an identity.
Everything below is one of those three. Recognising which one you are using is what turns memorised tricks into something you can improvise with.
Addition and subtraction
Round and compensate
47 + 38. Push 38 up to 40, add, then take the 2 back: 47 + 40 = 87, minus 2 = 85.
It scales. 486 + 297 becomes 486 + 300 = 786, minus 3 = 783.
Subtract by adding up
Borrowing is where mental subtraction falls apart, so avoid it. To do 83 − 47, count up from 47 instead: 47 + 3 gets you to 50, + 30 gets you to 80, + 3 gets you to 83. Total added: 3 + 30 + 3 = 36.
This is how shop assistants counted out change before tills did it, and it is faster than the written algorithm because nothing is ever carried.
Complements from a power of ten
1000 − 367. Subtract each digit from 9 and add 1 to the last: 9−3 = 6, 9−6 = 3, 10−7 = 3, giving 633. No borrowing at any point.
For a number like 2000 − 1486, do it against 1999 and add one back: 1999 − 1486 = 513, then + 1 = 514.
Multiplication
Decompose
7 × 68. Nobody should be attempting that as written. Take it as 7 × 70 minus 7 × 2: 490 − 14 = 476.
Halve one side, double the other
The product does not change. 14 × 35 is awkward; halve the 14 and double the 35 and it becomes 7 × 70 = 490.
The easy multipliers
| To multiply by | Do this | Example |
|---|---|---|
| 5 | halve, then ×10 | 86 × 5 → 43 × 10 = 430 |
| 25 | ×100, then ÷4 | 36 × 25 → 3600 ÷ 4 = 900 |
| 50 | ×100, then ÷2 | 74 × 50 → 7400 ÷ 2 = 3700 |
| 9 | ×10, then subtract the number | 47 × 9 → 470 − 47 = 423 |
| 99 | ×100, then subtract the number | 63 × 99 → 6300 − 63 = 6237 |
Multiplying a two-digit number by 11
Split the digits and put their sum in the middle. 43 × 11 → 4, then 4+3 = 7, then 3 → 473.
The part that gets left out of the version you were taught in a corridor: when the middle sum reaches ten or more, it carries. 87 × 11 → 8, then 8+7 = 15, then 7. The 1 from the 15 carries into the leading digit: 8 + 1 = 9, leaving 5 in the middle. Answer 957.
Test any shortcut on a carry case before you trust it. That is where the half-remembered versions break.
Squaring a number that ends in 5
Take the digits before the 5, multiply by the next whole number up, and stick 25 on the end.
65² → 6 × 7 = 42, then 25 → 4225.
85² → 8 × 9 = 72, then 25 → 7225.
Difference of squares
This is the one that looks like magic and is just algebra. When two numbers sit the same distance either side of a midpoint, their product is the midpoint squared minus the distance squared.
47 × 53: midpoint 50, distance 3. So 50² − 3² = 2500 − 9 = 2491.
38 × 42: midpoint 40, distance 2. So 1600 − 4 = 1596.
The condition matters. Both numbers must be equidistant from a whole-number midpoint, which means they must differ by an even amount. For 47 × 52 the midpoint is 49.5 and the method stops being worth it.
Percentages
Anchor on one percent
Find 1% by moving the decimal point two places, then scale. 18% of 250: one percent is 2.5, so eighteen of them is 2.5 × 18 = 45.
x% of y equals y% of x
This is the most under-used fact in everyday arithmetic. 4% of 75 looks unpleasant; 75% of 4 is obviously 3. They are the same number, because both are 0.04 × 75.
Whenever a percentage looks awkward, try flipping it. 16% of 25 becomes 25% of 16 = 4.
Tips and splits
15% is 10% plus half of 10%. On 62: 6.20 + 3.10 = 9.30. 20% is 10% doubled. If you can find 10% you can find most of the percentages anyone asks you for out loud.
Checking your work
Casting out nines
A two-second check that catches most slips. Reduce each number by summing its digits repeatedly until one digit remains, do the same operation on those reduced values, and compare with the reduced form of your answer.
Say you calculate 384 × 27 = 10,368.
- 384 → 3+8+4 = 15 → 1+5 = 6
- 27 → 2+7 = 9 → 0 (nines cast out)
- 6 × 0 = 0
- 10,368 → 1+0+3+6+8 = 18 → 0
They match, so the answer survives the check. Had you written 10,358, its digits reduce to 8, which does not match 0 — caught immediately.
Know its blind spots, because a check you trust incorrectly is worse than no check. Casting out nines cannot see transposed digits: write 10,386 instead of 10,368 and the digit sum is identical, so the test passes on a wrong answer. It is also blind to any error that happens to be a multiple of nine. It tells you when something is definitely wrong, never that something is definitely right.
Divisibility tests
Useful for simplifying fractions and for spotting whether a division will come out clean.
- By 3 — digit sum divisible by 3. 4,821 → 4+8+2+1 = 15, so yes (4,821 ÷ 3 = 1,607).
- By 4 — last two digits divisible by 4. 1,536 → 36, so yes (1,536 ÷ 4 = 384).
- By 8 — last three digits divisible by 8.
- By 6 — passes both the 2 and the 3 tests.
- By 11 — alternate adding and subtracting the digits; if the result is 0 or a multiple of 11, so is the number. 2,915 → 2 − 9 + 1 − 5 = −11, so yes (2,915 ÷ 11 = 265).
Estimate first, calculate second
Round every number to one significant figure, get an approximate answer, and only then do the real arithmetic. If the two disagree by an order of magnitude, you made a mistake in the second step. This is the habit that catches the decimal-point errors, and those are the expensive ones.
Turning methods into fluency
Knowing a method and being able to use it under mild pressure are separate skills, and only the second one is useful. The gap between them is closed by repetition, and repetition needs to be short enough that you actually do it.
A few things that make a practice habit hold:
Daily beats weekly. Retrieval speed decays between sessions. Six five-minute sessions do more than one thirty-minute session, and they are easier to schedule.
Alternate speed and accuracy. Training only for speed teaches you to guess. Training only for accuracy lets you take as long as you like, which does not build fluency. Run both, separately, so you can see which one is limiting you.
Watch response time, not just score. Score conflates speed and accuracy. Response time is the cleaner signal of whether the retrieval is getting automatic, and it is the number that moves first.
Practise your worst operation. Everyone has one — usually division, sometimes subtraction with borrowing. It is also the one you will avoid unless something schedules it for you.
Mental Math & Memory Games is built around that pattern. It covers addition, subtraction, multiplication and division with difficulty that climbs from simple warm-ups to harder rounds, and it varies the pressure deliberately: timed sprints reward speed, accuracy rounds reward getting it right, survival challenges run until you slip, and boss quizzes put a marker at the end of a stretch of work. A daily math challenge gives you a reason to open it on a day you had not planned a session.
Scores, accuracy, streaks, response time and personal bests are all recorded with a history you can look back through, which is what makes progress visible rather than a feeling. Alongside the drills there are techniques for percentages, estimation, number patterns and faster calculation, with games to apply each one straight away. On the memory side, number, grid, sequence and delayed-recall activities change what a session asks of you without changing how long it takes.
Sessions can be two minutes or a long training run, which is the property that makes daily practice realistic. It is free on Google Play, supported by ads, with optional in-app purchases, and it sits in the education and brain games category next to our history app — if you want a second habit with a different shape, the world history timeline guide covers how to use that one properly.
The short version
Mental arithmetic is a small number of ideas reused: round and compensate, decompose and recombine, swap in an easier equivalent. Learn those three and most published "tricks" become things you could have derived.
Check with casting out nines while remembering what it cannot see, estimate before you calculate so decimal errors surface early, and practise in short daily sessions rather than long occasional ones. The payoff is narrow and real — you get quicker and more accurate at arithmetic — and that is enough of a reason.
For the individual methods worked through in full, including the conditions where each one stops being exact, see mental math tricks that actually work.
Good to know
Frequently asked questions
Does mental math practice make you smarter?
What is the single most useful mental math technique?
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Do the multiplication shortcuts always work?
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Sources
- Casting Out Nines (opens in a new tab)Wolfram MathWorld — accessed