GCSE · Maths · AQA · Spec 8300
Inverse operations and order of operations
Quick: is 10 − 4 + 3 equal to 9 or 3? The rule you probably learnt can talk you into the wrong one. Let's find out why.
Order of operations
Which goes first?
Pick an expression, then choose the operation you'd do first. Commit before you check: the reason appears whether you're right or wrong.
Still to sort
Brackets (or a grouping sign) (0)
Also anything a root sign or a fraction line stretches right over.
Where the line is: A root sign or fraction line that covers a whole sum groups it, just like brackets. A root over a single number is just a root.
A power or a root (0)
Squares, cubes and square roots, once nothing grouped is left.
Where the line is: A power or root sticks to the number right next to it. In 5 × 2², only the 2 is squared.
The × (0)
Shares a level with ÷.
Where the line is: × goes first only when it is further left than every ÷, and nothing from a higher level is left.
The ÷ (0)
Shares a level with ×.
Where the line is: ÷ doesn't automatically beat ×. It goes first only when it is further left than every ×, and nothing from a higher level is left.
The + (0)
Shares a level with −.
Where the line is: + doesn't beat −. On their shared level, whichever is further left goes first.
The − (0)
Shares a level with +.
Where the line is: − goes first whenever it is the leftmost + or − still waiting.
Look at what caught you out. Usually it's the order of the letters, not the maths. × and ÷ share a level, and so do + and −.
WHAT YOU'VE LEARNED
A quick recap of today's lesson.
BIDMAS (or BODMAS) is a list of levels, not six steps in a row. And an operation followed by its inverse simply cancels.
What you need to know
- Work in levels: brackets first (including anything a root sign or fraction line covers), then powers and roots, then × and ÷, then + and −.
- × and ÷ are on the same level, so do whichever comes first reading left to right. The same goes for + and −.
- A power or root belongs to the number right next to it: in 2 × 3², only the 3 is squared, so 2 × 3² = 2 × 9 = 18.
- Inverse operations undo each other: × and ÷, + and −, squaring and square-rooting. Dividing by a number is the same as multiplying by its reciprocal.
- An operation followed by its inverse cancels, leaving what you started with: 58 × 6 ÷ 6 = 58.
The big picture
Operations follow an agreed order of levels. Brackets come first, and a root sign or fraction line that stretches over a sum groups it just like brackets. Next come powers and roots. Then × and ÷ together, worked left to right. Then + and − together, also worked left to right. Separately, every operation has an inverse that undoes it: × and ÷, + and −, squaring and square-rooting. When an operation is followed by its inverse, the pair cancels and you're left with what you started with. That can turn a calculation that looks like it needs a calculator into no calculation at all. And to undo a whole chain of operations, you use the inverses in reverse order.
Key points
Worked example
Problem
Work out 30 − √(12 + 13) × 2 + 4
⚠ Watch out
Treating the order as six steps in a strict line, so every + comes before any −. That makes 20 − 5 + 3 into 20 − 8 = 12. But + and − share a level, and the − is further left: 20 − 5 = 15, then 18.
Memory hook
Levels, then left to right. Picture the order as a staircase: × and ÷ share a step, + and − share a step, and on a shared step you read like a book. And for inverses: do it, undo it, and you're back where you began.
Check yourself
Without a calculator, work out 18 − 6 ÷ 3 + 1 and 250 × 13 ÷ 13. For each one, say which operation you did first and why. (Answers: 17 and 250.)
Flashcards
(6)What are the levels in the order of operations?
Two operations are on the same level. Which goes first?
What does a fraction line do when it sits under a whole sum?
What is an inverse operation?
An operation is followed by its inverse. What's left?
Can you cross a 3 out of the 6 in (x + 6) ÷ 3 to get x + 2?
Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.
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