Maths SolverThe internet is full of mental maths “tricks.” Some are genuinely useful and only take ten minutes to learn. Some are beautifully presented but in practice are slower than just doing the arithmetic the normal way. And some are completely made up — they happen to work for the cherry-picked example in the video and fail for anything else.
This is a personal short list. The criterion is simple: I will only list tricks I have actually used in front of a class without having to think about whether they apply. If a trick has too many edge cases to remember, the trick is worse than just doing the sum.
The five tricks worth learning
| Trick | Example | Why it works |
|---|---|---|
| Percentage swap | 4% of 75 = 75% of 4 = 3 |
$\tfrac{a}{100} \cdot b = \tfrac{b}{100} \cdot a$ |
| Multiply a 2-digit number by 11 | 11 × 36 = 3_(3+6)_6 = 396 |
$11n = 10n + n$, which spreads the digits with their sum in the middle |
| Square a number ending in 5 | 35² = (3·4)_25 = 1225 |
$(10a + 5)^2 = 100a(a+1) + 25$ |
| Multiply by 5 | 48 × 5 = (48 ÷ 2) × 10 = 240 |
$5 = 10 / 2$ |
| Subtract from 100 | 100 − 73 = 27 (each digit pair sums to 9, last to 10) |
The digits of $99 - n$ sum to 9; add 1 at the end |
These five cover most of the mental arithmetic you do in real life: working out tips, splitting bills, ballparking percentages on a price tag, doing fast estimates of how big a number is going to be. Let me walk through each one, because the table makes them look more cryptic than they are.
1. The percentage swap
This is the trick I use most often. Computing “$4\%$ of $75$” in your head is awkward; computing “$75\%$ of $4$” is trivial — it’s three quarters of $4$, which is $3$. The two expressions are mathematically identical, because both are $\tfrac{4 \cdot 75}{100}$. The trick is recognising that you can swap them whenever one of the two versions is easier.
A few real-world examples I have used in the last week:
- $8\%$ tip on a $\pounds 25$ bill: $25\%$ of $\pounds 8$ = $\pounds 2$.
- $15\%$ of $\pounds 60$: $60\%$ of $\pounds 15$ = $\pounds 9$ (or, easier: $10\% + 5\% = 6 + 3$).
- $3\%$ of $\pounds 200$: $200\%$ of $\pounds 3$ = $\pounds 6$.
You don’t need to memorise anything for this one. You just need to make a habit of asking, before you start computing $X\%$ of $Y$, whether $Y\%$ of $X$ would be easier.
2. Multiplying a two-digit number by 11
For $11 \times 36$, write the two digits with a gap in the middle ($3$ _ $6$) and put their sum in the middle: $3$, $9$, $6$, giving $396$. If the digit sum is more than $9$, you carry, just like normal addition: $11 \times 78$ gives $7$, $15$, $8$, which becomes $858$.
This works because $11n = 10n + n$, which is the same as adding the number to itself shifted left by one place. So for $36$:
$$11 \times 36 = 360 + 36 = 396.$$
Same arithmetic, different bookkeeping. Once you have done it five or six times the “split and add the digits” version becomes faster than the long multiplication.
3. Squaring a number ending in 5
For $35^2$: take the leading digit (3), multiply by the next integer (4), giving 12. Then stick “25” on the end: $1225$. For $75^2$: $7 \times 8 = 56$, then 25, giving $5625$. For $115^2$: $11 \times 12 = 132$, then 25, giving $13{,}225$.
The reason this works is a bit of algebra worth knowing on its own. Any number ending in 5 can be written as $10a + 5$, and squaring it gives $100a^2 + 100a + 25 = 100 \cdot a(a + 1) + 25$. The first part is “$a$ times the next integer, in the hundreds place”, which is exactly “$a(a + 1)$ followed by 25.”
This trick is genuinely fast and impressively reliable. Drop it in a casual conversation and people will think you are doing something exotic.
4. Multiplying by 5
$48 \times 5$: instead of multiplying, halve the number ($24$) and then multiply by $10$ (giving $240$). This works because $5$ is half of $10$, so anything you multiply by $5$ is the same as halving and adding a zero.
For odd numbers it gives a half: $37 \times 5$ becomes $18.5 \times 10 = 185$. You don’t even need the half — you can just do $37 \times 10 / 2 = 370 / 2 = 185$. Either order works.
5. Subtracting from 100 (or 1000)
For $100 - 73$: each digit except the last pairs up to make $9$ (so $9 - 7 = 2$); the last digit pairs up to make $10$ (so $10 - 3 = 7$). Answer: $27$. For $100 - 86$: $9 - 8 = 1$, $10 - 6 = 4$. Answer: $14$.
Same trick for $1000 - $ a 3-digit number. $1000 - 437$: $9 - 4 = 5$, $9 - 3 = 6$, $10 - 7 = 3$, giving $563$.
This is the trick I use when working out change at the till. Faster than long subtraction because there is no borrowing.
The tricks I would not bother learning
If you spend any time on mental-arithmetic YouTube, you will run into “Vedic maths,” “Trachtenberg,” and various “ancient” methods that promise to revolutionise your arithmetic. My honest take after trying most of them: they are mostly just regular arithmetic with a different notation, and the speed benefit only appears for very narrow classes of problem.
The “Vedic” method for multiplying two-digit numbers by two-digit numbers, for example, is a perfectly valid algorithm. It just is not faster than the standard one once you account for the mental load of remembering which special case you are in. For problems outside its sweet spot it is slower than long multiplication.
The handful of tricks above are different because each one identifies a specific algebraic identity that lets you avoid a step. Multiplying by $5$ via halving works because $5 = 10/2$ is genuinely shorter than multiplying digit by digit. The $11$ trick works because $11n$ has a specific structure. These are not just rebranded versions of normal arithmetic; they actually save effort.
What mental arithmetic is actually for
The point of mental arithmetic is not to impress people, and it isn’t to win calculator-free competitions. It is to:
- Sanity-check. When the calculator says $\pounds 4{,}320$ for the bill split between five people, you want a fast mental check that $5 \times 800$ is in the right ballpark.
- Skip the calculator. Lifting your phone to compute $25 + 30$ is a small humiliation. Picking it up to compute $20\%$ of $40$ is bigger.
- Build feel for numbers. Students who can do mental arithmetic have better intuition for whether an answer to an algebra problem is plausible. The skill transfers.
None of these applications need fast computation; they need roughly correct, fast enough computation. The five tricks above are good for that.
Practice for ten minutes a week, not an hour
The thing that actually makes mental arithmetic stick is using it in real life: working out the tip, ballparking the receipt before checking, splitting the bill at dinner. Doing five mental sums every day at the supermarket beats half an hour of mental-arithmetic drill on a Sunday, because the supermarket sums have stakes (you remember when you got it wrong) and the drills don’t.
If you want a small daily exercise: while you wait for the kettle, pick a two-digit number and square it using the trick above (if it ends in 5) or by writing it as $(a + b)^2$ in your head (if it doesn’t). Sixty seconds a day for a month and you will be embarrassingly faster at mental arithmetic than you are now. That is the entire training regime.