When there are no brackets
Brackets say which part of a line to work out first. When a line has no brackets, it still needs one meaning, so that everyone who reads it gets the same answer. The rule is: multiply and divide before you add and subtract.
So in 3 + 4 × 2, the multiplication goes first, even though it is written second: 4 × 2 = 8. Then 3 + 8 = 11.
There is a reason for the rule. 4 × 2 means two lots of 4. That is one amount, 8, and the 3 is added to that amount. Adding 3 + 4 first would pull one of the 4s away from the × 2 it belongs to.
3 + 4 × 2 is 3, and then two lots of 4: 3 + 8 = 11 squares.
this evaluates (2 + 3) × 4 = 20, which is not what 2 + 3 × 4 means
Apply the order of operations
The tiles show 2 + 3 × 4 read two ways. Adding first makes (2 + 3) × 4: 5 rows of 4, which is 20. Multiplying first keeps the 2 apart from 3 rows of 4, which is 2 + 12 = 14. Choose the reading the rule gives.
Dividing goes first too
Division is the same kind of step as multiplication, so it also comes before adding and subtracting. In 20 − 12 ÷ 4, work out 12 ÷ 4 first. Sharing 12 into 4 equal groups puts 3 in each, so 12 ÷ 4 = 3. Then 20 − 3 = 17.
Working from left to right instead gives 20 − 12 = 8, and then 8 ÷ 4 = 2. That divides the 8 by 4, but the line divides only the 12.
12 squares in 4 equal rows: each row holds 3, so 12 ÷ 4 = 3.
Then the 3 is taken away from 20: 20 − 3 = 17.
Adding and subtracting: left to right
Adding and subtracting share a rank: neither one comes before the other. When a line has only + and −, work from left to right. In 10 − 4 + 3, first 10 − 4 = 6, and then 6 + 3 = 9.
Adding first would give 4 + 3 = 7, and then 10 − 7 = 3. That takes the 3 away, but the line adds it.
Multiplying and dividing share a rank in the same way, so they also go from left to right: 8 ÷ 2 × 4 = 4 × 4 = 16. Multiplying first would give 8 ÷ 8 = 1, which divides by the 4 when the line multiplies by it.
÷ and × are one tier and + and − another: within a tier the first operation met is the first one done, left to right
Choose left to right and scan to the end
This opens on 8 − 2 + 5 with the adding done first, which takes 7 away and leaves 1. Choose left to right and slide to the end: 8 − 2 = 6, and 6 + 5 = 11. The other expression, 8 ÷ 2 × 4, works the same way.
The rule in order
With no brackets, work in two passes. First do every × and ÷, from left to right. Then do every + and −, from left to right.
In 18 − 2 × 5 + 12 ÷ 4, the first pass gives 2 × 5 = 10 and 12 ÷ 4 = 3, so the line becomes 18 − 10 + 3. The second pass gives 18 − 10 = 8, and then 8 + 3 = 11.
Worked example: One Item and Several of Another in One Total
Question A school bag costs $48 and a file costs $12. Mrs Tan buys 1 school bag and 6 files. (a) Write one expression for the total cost. (b) Find the total cost.
1.The bag costs $48. The files cost 6 lots of $12, which is 6 × 12.
There is one bag at $48 and there are six files at $12 each. 2.(a) One expression for the total cost is 48 + 6 × 12.
(a) The total cost is 48 + 6 × 12. 3.There are no brackets, so multiply before adding: 6 × 12 = 72.
Multiply before adding: 6 × 12 = 72. 4.(b) Now add: 48 + 72 = 120. The total cost is $120. Check: six files cost 12 + 12 + 12 + 12 + 12 + 12 = 72, and 72 + 48 = 120.
(b) 48 + 72 = 120. The total cost is $120.
Answer: (a) 48 + 6 × 12; (b) $120
Common mistakes
- Adding first because the addition is written first: 48 + 6 = 54, then 54 × 12 = 648. Multiplication is done before addition, wherever it is written in the line.
- Writing (48 + 6) × 12. These brackets would mean 54 things that each cost $12, but the bag does not cost $12.
Worked example: One Expression, Two Pupils, Two Answers
Question Ali and Bea both work out 36 − 12 ÷ 4 + 2. Ali gets 8 and Bea gets 35. (a) What is the correct value? (b) Who is right, and how did the other pupil get a different answer?
1.There are no brackets, so the division comes first: 12 ÷ 4 = 3. The expression becomes 36 − 3 + 2.
There are no brackets, so the division comes first: 12 ÷ 4 = 3. 2.Only subtraction and addition are left, so work from left to right: 36 − 3 = 33.
Only subtraction and addition are left. Work from left to right: 36 − 3 = 33. 3.(a) Then 33 + 2 = 35. The correct value is 35.
(a) 33 + 2 = 35. The correct value is 35. 4.(b) Bea is right. Ali worked from left to right from the start: 36 − 12 = 24, then 24 ÷ 4 = 6, then 6 + 2 = 8. He subtracted before he divided.
(b) Bea is right. Ali subtracted before he divided: 36 − 12 = 24, 24 ÷ 4 = 6, 6 + 2 = 8.
Answer: (a) 35; (b) Bea is right. Ali worked from left to right and got 8
Common mistakes
- Adding before subtracting in 36 − 3 + 2: 3 + 2 = 5, then 36 − 5 = 31. Subtraction and addition are done in the order they appear, so the 3 is subtracted and then the 2 is added.
- Working the whole expression from left to right, as Ali did. Division is done before subtraction even when the subtraction is written first.