Maths › The four operations
The four operations
Adding, subtracting, multiplying and dividing are the tools every other topic borrows. Here we tidy up the written methods, add a few quick mental ones, and learn the order to use them in when several turn up in the same sum.
What these questions are
This is the workhorse page. The maths itself is familiar — you have been adding and subtracting since Year 1 — so the value here is in tidy, reliable methods: setting your working out clearly, knowing which order to do things in, and having a quick way to check your answer is right.
On this page you’ll cover three things:
- Written & mental methods — the column methods for all four operations, including with large numbers, plus a few quick mental strategies.
- Order of operations & brackets — what to do first when a sum mixes several operations together, and how brackets change that order.
- Function machines & inverse operations — following a chain of operations through, and running it backwards to find a missing input.
How to crack them
- Line up your columns by place value — ones under ones, tens under tens — or the exchanging will go wrong.
- Estimate first, so you’d notice straight away if your exact answer came out a silly size.
- In a string of operations, × and ÷ always happen before + and −.
- When you’re not sure, check your answer with the inverse operation.
A little tip: we’ve bolded the key instruction word in each question — like altogether, left over, share or each — so you always know exactly what you’re being asked to find.
1. Written & mental methods
Each of the four operations has a reliable written method — a way of setting it out so the answer falls out step by step, without needing to hold the whole calculation in your head at once.
They all lean on the same idea: line the digits up by place value, then work through the columns in order.
Addition and subtraction
For column addition, add each column starting from the ones, on the right. Whenever a column adds up to 10 or more, write down the ones digit and exchange ten of them for one extra in the column to the left — this is often called “carrying”.
Column subtraction works the same way in reverse. If the top digit in a column is smaller than the one you’re taking away, exchange one from the column to its left first — “borrowing”. If that column is a zero, the exchange has to reach one column further left still.
Multiplication and division
For long multiplication, split the second number by place value, multiply the first number by each part separately, then add the results together. The worked example below shows exactly this.
For division, short division handles dividing by a single digit; long division handles bigger divisors the same way, just with more room to show the working.
Either can leave a remainder — what’s left over once the dividing number no longer fits exactly. What you do with it depends on the question: sometimes you report it, sometimes you round the answer up or down instead.
Every one of these methods works exactly the same way for large numbers too. Nothing new is needed — just more columns.
Mental strategies
For friendlier numbers, it’s often quicker to work in your head using one of these:
- Partitioning — split a number into friendly parts and deal with each separately, e.g. 47 + 35 = 47 + 30 + 5.
- Near-doubles — if two numbers are almost the same, double one and adjust, e.g. 58 + 59 = 58 + 58 + 1.
- Rounding and adjusting — round a number to make the sum easy, then correct for the rounding afterwards, e.g. 99 + 35 = 100 + 35 − 1.
- Split the second number by place value. 26 is 20 + 6, so the sum splits into two friendlier parts.
- Multiply by each part. 34 × 20 = 680, and 34 × 6 = 204.
- Add the two results together. 680 + 204 = 884.
- Check with an estimate. 34 rounds to 30 and 26 rounds to 30, so 30 × 30 = 900 — close to 884, which shows the exact answer is a sensible size.
Your go: written & mental methods
A mix of adding, subtracting, multiplying and dividing — read carefully to see which is needed, then press an answer.
2. Order of operations & brackets
When a sum mixes several operations together, the order you do them in changes the answer — so everyone has to agree on the same order. That order is often remembered as BODMAS or BIDMAS: Brackets first, then Orders/Indices (powers, which you’ll meet on another page), then Division and Multiplication, then Addition and Subtraction.
Division and multiplication are equal partners — do whichever comes first, reading left to right. The same goes for addition and subtraction. The one rule that overrides everything is brackets: whatever sits inside them is always done first, no matter what it looks like from outside.
Same digits, different order
- Without brackets, multiply first. 3 + 4 × 5: do 4 × 5 = 20 first, then 3 + 20 = 23.
- With brackets, the bracket goes first instead. (3 + 4) × 5: do 3 + 4 = 7 first, then 7 × 5 = 35.
- Same three numbers, same two operations — a different order gives a different answer. That is the whole reason brackets exist.
Division before subtraction
- Scan for × or ÷ first. There is a ÷ in this sum, so it happens before the −.
- Work out the division. 6 ÷ 2 = 3.
- Now do the subtraction. 20 − 3 = 17.
Your go: order of operations & brackets
Work out the value of each expression — remember, brackets (where there are any) always come first.
3. Function machines & inverse operations
A function machine is just a diagram for a chain of operations. A number goes in, each box does something to it in turn, and a number comes out. To find the output, do whatever comes first exactly as it’s written, box by box, left to right.
Sometimes you’re given the output instead, and asked to find the input. To do that, run the machine backwards: start from the output, work right to left, and undo each box using its inverse — the operation that reverses it.
The inverse of + is −, the inverse of − is +, the inverse of × is ÷, and the inverse of ÷ is ×.
Inverses are also a ready-made way to check an ordinary answer: if a sum is right, reversing it with the inverse operation should always take you back to where you started.
Forward: 5 → ×3 → 15 → +2 → 17. Reverse: 17 → −2 → 15 → ÷3 → 5, using the inverse of each step in reverse order.
- Forwards, apply the steps in order. 5 × 3 = 15, then 15 + 2 = 17. So an input of 5 gives an output of 17.
- Backwards, undo the steps in reverse order, using inverses. Start from 17 and undo the last step first: the inverse of +2 is −2, so 17 − 2 = 15.
- Undo the first step last. The inverse of ×3 is ÷3, so 15 ÷ 3 = 5 — back to the original input.
Your go: function machines & inverse operations
Some questions give the input and ask for the output. Others give the output and ask you to work backwards for the input — remember to reverse the order as well as the operations.
That is the four operations
Solid written methods, the right order, and a quick way to check your working — these turn up on every other page from here. When you’re ready, head back and try another topic.
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