Before this page: be able to read a base and its index from index notation and powers, and work out a root of a perfect square from square roots.
Does 3 + 4 × 5 come to 23 or to 35?
Adding first leaves 7 × 5, which is 35; multiplying first leaves 3 + 20, which is 23. Neither working has a slip in it; the answers differ only in which was done first.
By agreement, multiplying and dividing are done before adding and subtracting, so 3 + 4 × 5 = 3 + 20 = 23.
Three operations allow more orders, and 2 + 3 × 4 − 1 comes to four different answers depending on which is worked first.
2 + 3 × 4 − 1
left to right
19
right to left
11
the ends first
15
the × first
13
The agreed order, the × first, gives 13. A cake at £5 and three coffees at £4 cost 5 + 3 × 4 = 17 pounds. Gathering 5 + 3 first would charge £32.
Example: work out 10 − 2 × 3
Take 2 × 3 from 10.
- The terms are 10 and 2 × 3. The minus sign separates them.
- 2 × 3 = 6 Two threes make one amount of 6.
- 10 − 2 × 3 = 4 The 6 stands where 2 × 3 stood: 10 − 6 = 4.
Brackets make their contents one amount
Brackets are how to ask for the 3 + 4 to be done first:
3 + 4 × 5 and (3 + 4) × 5
A rectangle 4 by 5
4 × 5 is a rectangle: four squares up the side, five along the bottom, 20 in all.
A pair of brackets gathers its contents into one amount before anything outside is worked. In 2 × 3 + 4 × 5, the bracket placement decides which numbers meet first.
2 × 3 + 4 × 5 = 26
Brackets round either product in 2 × 3 + 4 × 5 change nothing, because each product is gathered before the adding anyway.
Example: insert brackets in 3 + 5 × 2 to make 16
Find the placing that gives 16.
- 3 + 5 × 2 = 13 Five twos make one amount of 10, and 3 + 10 = 13.
- (3 + 5) × 2 = 16 The 2 now multiplies 3 + 5 rather than the 5 alone: eight taken twice.
Powers and roots before the multiplying around them
A power is worked before the multiplying around it: 3 × 42 is three sixteens, 48. A square root is worked at the same stage: 5 × 9 = 15.
2 + 3 × 42 ÷ 8
The power first
4 × 4 = 16 is worked before the × and the ÷ either side of it.
2 + 3 × 42 ÷ 8 = 8
- Brackets first, and the innermost pair before the pair around it.
- Then powers and roots.
- Then multiplying and dividing, from the left.
- Then adding and subtracting, from the left.
BIDMAS spells the same order in six letters: brackets, indices, division, multiplication, addition, subtraction. The D and M are one strength and so are the A and S: D before M does not mean divide before multiply, and A before S does not mean add before subtract.
Example: work out 40 ÷ 5 × 2
Divide 40 by 5, then multiply by 2.
- 40 ÷ 5 = 8 Dividing and multiplying are one strength, so the leftmost of the two comes first.
- 40 ÷ 5 × 2 = 16 Eight doubled. Multiplying first would give 40 ÷ 10 = 4, a different calculation.
Bars that group without brackets
A root's bar fixes how far the root reaches. 9 + 16 has both numbers under one bar, so the 9 + 16 is added before the root is taken, and the answer is 5. Shortening the bar to cover the 9 alone leaves the 16 outside it, and 9 + 16 = 19.
A fraction bar groups the same way, one amount above it and one below, so 12 + 85 − 1 is 20 ÷ 4, which is 5.
Example: work out 7 + 59 − 3
Divide 7 + 5 by 9 − 3.
- 7 + 5 = 12 The bar groups 7 + 5 as one amount.
- 9 − 3 = 6 The bar groups 9 − 3 the same way.
- 12 ÷ 6 = 2 The bar divides one amount by the other.
Common mistakes
- Working straight from the left 2 + 6 × 3 is not 24. The 6 × 3 is worked before the addition, so the amount added to 2 is 18: 2 + 6 × 3 = 2 + 18 = 20.
- Multiplying before the index is applied In 2 × 52 the index sits on the 5 alone, so the 5 is squared and the 2 multiplies the 25: 2 × 25 = 50, not 100.
- Rooting a sum term by term One bar covers all of 36 + 64, so the number under it is 100 and 36 + 64 = 10, not 14.
- Adding before subtracting 15 − 6 + 4 is not 5. Adding and subtracting have the same strength, so the 15 − 6 on the left is worked first: 9 + 4 = 13.
Summary
- By convention multiplying and dividing come before adding and subtracting: 3 + 4 × 5 = 3 + 20 = 23.
- Brackets are how to ask for a different grouping, so (3 + 4) × 5 = 35.
- Powers and roots are worked before the multiplying around them: 3 × 42 = 48.
- Operations of the same strength are worked from the left: 40 ÷ 5 × 2 = 16.
- A root's bar and a fraction bar group what is under them, so 9 + 16 = 5.