Order of Operations

Learn the order to do a calculation in, put brackets in to make a statement true, and type fractions into a calculator correctly.

In the exam for Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024, Edexcel IGCSE Maths 4MA1 · 48 real past-paper questions below

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BIDMAS: Which Part Goes First

A calculation with more than one operation has a fixed order. Everyone follows it, so everyone gets the same answer.

Order	What you do
1. Brackets	anything inside ( )
2. Indices	powers and roots
3. Divide and Multiply	left to right
4. Add and Subtract	left to right

Worked example

1
$2 + 3 \times 4$ — There is an add and a multiply. Multiply comes before add, so do 3 × 4 first.
2
$= 2 + 12$ — 3 × 4 = 12. The 2 waits until the multiply is done.
3
$= 14$ — Only the add is left: 2 + 12 = 14.
4
$(2 + 3) \times 4$ — Same numbers, now with brackets. Brackets come first of all, so 2 + 3 is done before the multiply.
5
$= 5 \times 4$ — 2 + 3 = 5.
6
$= 20$ — 5 × 4 = 20. The brackets changed the answer from 14 to 20.

Key idea

In every worked example here, the coloured part is the part you do first.

Key idea

Puzzle: make 7 using exactly four 4s. One answer: $44 \div 4 - 4$ . Divide first, $44 \div 4 = 11$ , then $11 - 4 = 7$ . Now make 9 with four 4s. (One answer: $4 + 4 + 4 \div 4$ .)

Work Out Without a Calculator

The 1-mark question: work out a line like $18 \div 3 + 2 \times 7$ . Do one level at a time and write each line.

Worked example

1
$18 \div 3 + 2 \times 7$ — No brackets or powers here. Divide and multiply come before add, so do both of them first.
2
$= 6 + 14$ — 18 ÷ 3 = 6 and 2 × 7 = 14. Write this line down: it shows the order you used.
3
$= 20$ — Only the add is left: 6 + 14 = 20.

With powers, do the powers before anything else outside brackets.

Worked example

1
$4^{2} + 2^{3} \times 3$ — No brackets, so powers come first. Work out both powers before anything else.
2
$= 16 + 8 \times 3$ — 4² = 4 × 4 = 16 and 2³ = 2 × 2 × 2 = 8. Next comes the multiply, because it is before add.
3
$= 16 + 24$ — 8 × 3 = 24.
4
$= 40$ — Add last: 16 + 24 = 40.

Rule

Divide and multiply are equal partners. So are add and subtract. For partners, go left to right: $12 \div 4 \times 3 = 3 \times 3 = 9$ .

Common mistake

Going left to right through everything: $18 \div 3 + 2 = 8$ , then $8 \times 7 = 56$ . Multiply and divide come before adding.

Key idea

When a question says “show all your working”, the middle line, like $6 + 14$ , earns a mark on its own. Write it even if you can do it in your head.

Negative Numbers in BIDMAS

Exams often put negative numbers into a BIDMAS question. The order does not change. Keep each negative number in brackets so its sign stays with it.

Worked example

1
$7 - 4 \times (- 3)$ — Multiply comes before subtract, so do 4 × (−3) first.
2
$4 \times (- 3) = - 12$ — A positive times a negative gives a negative.
3
$7 - (- 12)$ — Put the answer back in its place, in brackets, so its minus sign is not lost.
4
$- (- 12) = + 12$ — Taking away a negative is the same as adding.
5
$7 + 12 = 19$ — Now add: 7 + 12 = 19.

Two negatives multiplied. Do every multiply first, then add.

Worked example

1
$- 5 \times (- 2) + 3 \times (- 4)$ — Two multiplies and an add. Both multiplies go first.
2
$- 5 \times (- 2) = 10$ — A negative times a negative gives a positive.
3
$3 \times (- 4) = - 12$ — A positive times a negative gives a negative.
4
$10 + (- 12) = 10 - 12$ — Put both answers back. Adding a negative is the same as taking away.
5
$= - 2$ — 12 is bigger than 10, so the answer is below zero: −2.

A power and a minus sign. Look closely at what the power is attached to.

With a bracket

(- 3)^{2} = (- 3) \times (- 3) = 9

The whole bracket is squared, minus sign and all.

No bracket

- 3^{2} = - (3 \times 3) = - 9

Only the 3 is squared. The power comes before the minus.

Common mistake

Typing −3² into a calculator and expecting 9. The calculator squares the 3 first and shows −9. Type $(- 3)^{2}$ with the bracket.

Put In Brackets to Make It True

The 1-mark question: write one pair of brackets so the answer is correct. Try each place, and check.

Worked example

1
$4 + 2 \times 3 = 18 ?$ — The question. You need one pair of brackets that makes the left side equal 18.
2
$4 + 2 \times 3$ — First see what it gives with no brackets. Multiply comes before add, so do 2 × 3.
3
$= 4 + 6 = 10$ — 2 × 3 = 6, and 4 + 6 = 10. That is not 18, so brackets are needed.
4
$(4 + 2) \times 3$ — Try brackets round the add, so the add happens before the multiply.
5
$= 6 \times 3$ — 4 + 2 = 6.
6
$= 18$ — 6 × 3 = 18. It matches, so the brackets go round 4 + 2.

Harder ones hide the brackets at the end.

Worked example

1
$50 - 12 \div 6 - 2 = 47 ?$ — The question. Find where one pair of brackets makes it 47.
2
$50 - 12 \div 6 - 2$ — First try no brackets. Divide comes before subtract, so do 12 ÷ 6.
3
$= 50 - 2 - 2 = 46$ — 12 ÷ 6 = 2. Then subtract left to right: 48, then 46. Not 47, so brackets are needed.
4
$50 - 12 \div (6 - 2)$ — Try brackets round the last part, so 6 − 2 happens before the divide.
5
$= 50 - 12 \div 4$ — 6 − 2 = 4.
6
$= 50 - 3$ — 12 ÷ 4 = 3.
7
$= 47$ — 50 − 3 = 47. It matches.

A row of minus signs. When there are only + and −, a bracket straight after a minus changes the answer.

Worked example

1
$12 - 5 - 3 + 4 = 14 ?$ — The question. With only add and subtract and no brackets, you work left to right.
2
$12 - 5 - 3 + 4 = 8$ — Left to right: 12 − 5 = 7, then 7 − 3 = 4, then 4 + 4 = 8. Not 14, so brackets are needed.
3
$12 - (5 - 3) + 4$ — Put brackets round 5 − 3. Now you take away just the 2 that is left, not 5 and then 3.
4
$= 12 - 2 + 4$ — 5 − 3 = 2.
5
$= 10 + 4$ — Left to right: 12 − 2 = 10.
6
$= 14$ — 10 + 4 = 14. It matches.

Common mistake

Not checking, or using two pairs. When the question says one pair, use exactly one: an extra pair loses the mark even if the answer is right. Then work out your version with BIDMAS. It must give the answer in the question.

Fractions on a Calculator

The 2-mark question: work out a fraction like this and write all the figures on the display.

\frac{7.3 + 2. 6 ^{2}}{3.1 \times 1.4}

The fraction line is a hidden bracket. Work out the whole top and the whole bottom, then divide.

Worked example

1
$top: 7.3 + 2. 6^{2}$ — Start with the whole top. Powers come before add, so square 2.6 first.
2
$= 7.3 + 6.76$ — 2.6² = 2.6 × 2.6 = 6.76.
3
$= 14.06$ — 7.3 + 6.76 = 14.06. This is the whole top.
4
$bottom: 3.1 \times 1.4 = 4.34$ — Now the whole bottom. It has only a multiply.
5
$14.06 \div 4.34$ — The fraction line means top divided by bottom. Do it last, once both are finished.
6
$= 3.239631336$ — Write every figure on the display. The question asks for all of them, so do not round.

Common mistake

Typing it in one line with no brackets. The calculator then divides only $2. 6^{2}$ by 3.1 and gives 10.35..., which is wrong. Put brackets round the top and round the bottom.

Key idea

A square root sign is a hidden bracket too: $16 + 9 = 25 = 5$ , not 4 + 9.

A root over the top, to 2 decimal places. Some questions put a root over the whole top and ask you to round.

\frac{14.2 + 1. 8 ^{3}}{2.5 \times 0.9}

Worked example

1
$top: 14.2 + 1. 8^{3}$ — The root covers everything under it, like a bracket. Work out the inside first: power before add.
2
$= 14.2 + 5.832$ — 1.8³ = 1.8 × 1.8 × 1.8 = 5.832.
3
$= 20.032$ — 14.2 + 5.832 = 20.032. This is everything under the root.
4
$20.032 = 4.47571...$ — Now take the square root of the whole inside. Keep all the figures; do not round yet.
5
$bottom: 2.5 \times 0.9 = 2.25$ — Work out the whole bottom.
6
$4.47571... \div 2.25 = 1.98920...$ — Top divided by bottom, using the full value of the top.
7
$\approx 1.99$ — Round to 2 decimal places. The third decimal place is 9, so round up.

Key idea

Write the full display, like 1.98920..., before you round. If your rounding slips, that line can still earn a mark.

Missing Numbers and Spotting Mistakes

The 1-mark question: fill the box so the calculation is correct. Work backwards, one operation at a time.

Worked example

1
$20 - □ \times 3 = 8$ — BIDMAS does the multiply before the take-away, so treat □ × 3 as one block.
2
$20 - 12 = 8$ — 20 take away what makes 8? The block must be 12, because 20 − 12 = 8.
3
$□ \times 3 = 12$ — So the box times 3 is 12.
4
$□ = 4$ — Undo the multiply: 12 ÷ 3 = 4. Check: 20 − 4 × 3 = 20 − 12 = 8.

Worked example

1
$(2 + □) \times 4 = 36$ — The bracket is worked out first, so treat it as one block that is multiplied by 4.
2
$(2 + □) = 9$ — Undo the multiply: 36 ÷ 4 = 9. So the bracket is worth 9.
3
$□ = 7$ — Undo the add: 9 − 2 = 7. Check: (2 + 7) × 4 = 9 × 4 = 36.

Missing signs. Some questions ask you to put +, − and × between the numbers, each once. Try the × in different places and check with BIDMAS.

Worked example

1
$4 □ 3 □ 5 □ 2 = 17$ — The question. A multiply makes numbers grow fastest, so decide where the × goes first.
2
$4 + 3 \times 5 - 2$ — Try × between 3 and 5, with + and − in the other gaps. BIDMAS does the multiply first.
3
$= 4 + 15 - 2$ — 3 × 5 = 15.
4
$= 19 - 2$ — Left to right: 4 + 15 = 19.
5
$= 17$ — 19 − 2 = 17. It matches, so the signs are +, × and −.

The other 1-mark question: explain what someone did wrong. Leo wrote $4 + 3^{2} = 49$ .

Worked example

1
$4 + 3^{2} \neq = 7^{2}$ — Leo added 4 + 3 first to get 7, then squared it to get 49.
2
$4 + 3^{2}$ — Powers come before add, so the 3 must be squared first.
3
$= 4 + 9$ — 3² = 3 × 3 = 9.
4
$= 13$ — 4 + 9 = 13, not 49.

Rule

Say which step should come first: Leo should have squared 3 first, then added 4.

Which Method? Read the Question

The wording tells you which kind of question it is.

You see	You do
Work out (no calculator)	BIDMAS, one level per line
Write brackets to make it correct	try each place, then check
A fraction, write all the figures	top and bottom first, or brackets
Fill in the missing number	work backwards
Explain what went wrong	say which step should be first
Negative numbers inside	same order; keep each negative in brackets
Round or check the display	Using a Calculator

Key idea

Always check your final line with BIDMAS. It takes ten seconds and catches most slips.

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Real past-paper questions on order of operations

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