With brackets
In (5 + 3) × 2, the brackets come first: 5 + 3 = 8. The bracket is now one number, and the line becomes 8 × 2 = 16. The × 2 doubles the whole of 5 + 3.
(5 + 3) × 2 is two lots of 5 + 3: 5 + 3 + 5 + 3 = 16 squares.
Without brackets
Take the brackets away and the line is 5 + 3 × 2. Now there is no bracket to go first, so the multiplication does: 3 × 2 = 6, and then 5 + 6 = 11. The × 2 doubles only the 3.
The numbers and the signs are the same in both lines. The brackets alone changed the answer from 16 to 11.
5 + 3 × 2 is 5, and then two lots of 3: 5 + 6 = 11 squares.
The order of operations
Put the rules together and they make one order, called the order of operations. First, work out anything in brackets. Next, do every × and ÷, from left to right. Last, do every + and −, from left to right.
Here are all four operations in one line: (10 − 4) ÷ 2 + 1. The bracket comes first: 10 − 4 = 6, so the line becomes 6 ÷ 2 + 1. Division comes before addition: 6 ÷ 2 = 3, so the line becomes 3 + 1. Last, add: 3 + 1 = 4.
The bracket leaves 6. Split into 2 equal rows, each row holds 3, so 6 ÷ 2 = 3, and 1 more makes 4.
A longer line
In 20 − 2 × (3 + 4), the bracket comes first, even though it is written last: 3 + 4 = 7. Then multiply: 2 × 7 = 14. Last, subtract: 20 − 14 = 6.
To check the order you used, write a bracket around each step as you do it: 20 − (2 × (3 + 4)). Each pair of brackets holds one step, and the innermost is done first.
Where the brackets make a difference
Leaving the brackets out changes the answer. Without them, (10 − 4) ÷ 2 + 1 becomes 10 − 4 ÷ 2 + 1, and the division goes first: 4 ÷ 2 = 2, so the line is 10 − 2 + 1 = 9, not 4. Only the 4 was divided, when the brackets divide the whole 6.
Brackets around a multiplication change nothing: 5 + (3 × 2) = 11, the same as 5 + 3 × 2, because the × was going first anyway. Brackets can change the answer only when they move a step earlier than the order of operations would put it.
Worked example: Large and Small Boxes, Some Removed, the Rest Repacked
Question A fruit seller has 5 large boxes with 24 apples in each and 8 small boxes with 12 apples in each. He removes 36 bad apples and packs the rest into bags of 6. (a) How many good apples are there? (b) Write one expression for the number of bags, and find its value.
1.The large boxes hold 5 × 24 = 120 apples and the small boxes hold 8 × 12 = 96 apples.
The large boxes hold 5 × 24 = 120 apples and the small boxes hold 8 × 12 = 96. 2.(a) Add the two kinds of boxes and take away the bad apples: 120 + 96 − 36 = 180. There are 180 good apples.
(a) 120 + 96 − 36 = 180 good apples. 3.All the good apples are packed, so their expression goes inside brackets before the division: (5 × 24 + 8 × 12 − 36) ÷ 6.
All the good apples are packed, so their expression goes inside brackets: (5 × 24 + 8 × 12 − 36) ÷ 6. 4.(b) The brackets are worth 180, so 180 ÷ 6 = 30. He packs 30 bags. Check: 30 × 6 = 180.
(b) The brackets are worth 180, and 180 ÷ 6 = 30 bags.
Answer: (a) 180 apples; (b) (5 × 24 + 8 × 12 − 36) ÷ 6 = 30 bags
Common mistakes
- Leaving out the brackets: 5 × 24 + 8 × 12 − 36 ÷ 6 = 120 + 96 − 6 = 210. Without brackets only the 36 bad apples are divided by 6.
- Adding the numbers of boxes first, 5 + 8 = 13, and multiplying by one box size. The large and small boxes hold different numbers of apples, so each kind is multiplied on its own.
Worked example: One Pair of Brackets to Make a Statement True
Question The statement 4 + 6 × 7 − 2 = 34 is not true. One pair of brackets makes it true. (a) What is the value of 4 + 6 × 7 − 2 with no brackets? (b) Where do the brackets go?
1.(a) With no brackets, multiply first: 6 × 7 = 42. Then 4 + 42 − 2 = 44, which is not 34.
(a) With no brackets, multiply first: 6 × 7 = 42, then 4 + 42 − 2 = 44. That is not 34. 2.Try the brackets round 4 + 6: (4 + 6) × 7 − 2 = 10 × 7 − 2 = 68. That is too large.
Brackets round 4 + 6 give 10 × 7 − 2 = 68. That is too large. 3.Try the brackets round 7 − 2: 4 + 6 × (7 − 2) = 4 + 6 × 5.
Brackets round 7 − 2 give 4 + 6 × 5. 4.(b) Multiply, then add: 6 × 5 = 30 and 4 + 30 = 34. The brackets go round 7 − 2, and the true statement is 4 + 6 × (7 − 2) = 34.
(b) 6 × 5 = 30 and 4 + 30 = 34. The statement is true as 4 + 6 × (7 − 2) = 34.
Answer: (a) 44; (b) 4 + 6 × (7 − 2) = 34
Common mistakes
- Putting the brackets round 6 × 7. The multiplication is already done first, so the value stays 44.
- After writing 4 + 6 × 5, adding first to get 10 × 5 = 50. The brackets are finished, but multiplication still comes before addition.