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Year 6 Maths Number Sentences & Order of Operations

Order of operations & grouping symbols - Two operations

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Theory

When a number sentence has more than one operation, everyone must do them in the same order to get the same answer. Multiply and divide before you add and subtract. When the operations are equal in rank (only \(+\) and \(-\), or only \(\times\) and \(\div\)), work from left to right. This Year 6 maths page shows how to use these rules with two operations and how to write number sentences for word problems.

An operation is \(+\), \(-\), \(\times\) or \(\div\). The order of operations is the agreed order for working them out.

Multiplication and division come before addition and subtraction, wherever they are in the line. In \(9 + 2 \times 7\), do \(2 \times 7\) first.

Multiplication and division have equal rank, and so do addition and subtraction. Operations of equal rank are worked from left to right: \(18 - 7 + 4 = 11 + 4 = 15\).

Brackets change the order: anything in brackets is worked out first. (Brackets are covered fully on the brackets page.)

The order of operations Three boxes in order: first brackets, work inside first; second multiply and divide, left to right; third add and subtract, left to right. 1 Brackets ( )work inside first 2 × and ÷left to right 3 + and −left to right
Work from the top box down. Inside a box, go from left to right.
Working out 35 minus 21 divided by 7 The line 35 − 21 ÷ 7 with 21 ÷ 7 highlighted, then 35 − 3 highlighted, then the answer 32. 35 − 21 ÷ 7 divide first 35 − 3 then subtract 32
In \(35 - 21 \div 7\) the division is done first.
Working out 18 minus 7 plus 4 The line 18 − 7 + 4 with 18 − 7 highlighted first, then 11 + 4, then the answer 15. 18 − 7 + 4 equal rank: left to right 11 + 4 then add 15
Only \(+\) and \(-\): work from left to right.

The rules for two operations:

\[ \times, \div \;\text{ before }\; +, - \]
× and ÷ before + and −
\[ \text{equal rank} \Rightarrow \text{left to right} \]
equal rank left to right
\[ 9 + 2 \times 7 = 9 + 14 = 23 \]
9+2×7=9+14=23
Not always left to right. In \(4 + 5 \times 6\) the multiplication is done first, even though it is second in the line.
Left to right for equal rank. \(48 \div 4 \div 2 = 12 \div 2 = 6\), not \(48 \div 2 = 24\).

Working out a number sentence with two operations

  1. Look for \(\times\) or \(\div\). If there is one (and a \(+\) or \(-\)), do it first.
  2. If both operations are the same rank, work from left to right.
  3. Rewrite the line after each step so you can see what is left.
Example 1 — Multiply first
Work out \(9 + 2 \times 7\).
Solution
\(9 + 2 \times 7\)\(=\)\(9 + 14\)
\(9 + 14\)\(=\)\(23\)
9+2×7=9+14=23

\(23\).

Example 2 — Multiply first (again)
Work out \(8 \times 6 - 10\).
Solution
\(8 \times 6 - 10\)\(=\)\(48 - 10\)
\(48 - 10\)\(=\)\(38\)
48−10=38

\(38\).

Example 3 — Divide first
Work out \(35 - 21 \div 7\).
Solution
\(35 - 21 \div 7\)\(=\)\(35 - 3\)
\(35 - 3\)\(=\)\(32\)
35−3=32

\(32\), as in the second diagram.

Example 4 — Left to right
Work out \(18 - 7 + 4\).
Solution
\(18 - 7 + 4\)\(=\)\(11 + 4\)
\(11 + 4\)\(=\)\(15\)
18−7+4=15

\(15\). (Adding first would give \(7\), which is wrong.)

Example 5 — Three to try
Find the value of each.
(i) \(11 + 4 \times 9\)
(ii) \(56 \div 8 + 3\)
(iii) \(90 - 5 \times 12\)
Solution
\(11 + 36\)\(=\)\(47\)
\(7 + 3\)\(=\)\(10\)
\(90 - 60\)\(=\)\(30\)
47,10,30

(i) \(47\)
(ii) \(10\)
(iii) \(30\)

Example 6 — Two divisions
Work out \(48 \div 4 \div 2\).
Solution
\(48 \div 4 \div 2\)\(=\)\(12 \div 2\)
\(12 \div 2\)\(=\)\(6\)
48÷4÷2=6

\(6\).

Example 7 — Write a number sentence
Mo buys \(6\) packs of \(8\) bread rolls and \(5\) single rolls. Write a number sentence for the number of rolls and work it out.
Solution
\(6 \times 8 + 5\)\(=\)\(48 + 5\)
\(48 + 5\)\(=\)\(53\)
6×8+5=53

\(6 \times 8 + 5 = 53\) rolls. The order rules mean no brackets are needed.

Example 8 — Find the mistake
Jack worked out \(4 + 5 \times 6\) from left to right and wrote \(54\). What is right?
Solution
\(4 + 5 \times 6\)\(=\)\(4 + 30\)
\(4 + 30\)\(=\)\(34\)
4+5×6=34

He added first. Multiplication comes first, so the answer is \(34\).

Common pitfalls

Always working left to right. \(4 + 5 \times 6 = 34\), not \(54\).
Adding before subtracting. \(18 - 7 + 4 = 15\). Equal-rank operations go left to right.
Dividing the wrong pair. \(48 \div 4 \div 2 = 6\), not \(24\).
Doing two steps at once. Rewrite the line after each step.

Frequently asked questions

What is the order of operations?

The agreed order: brackets first, then multiplication and division, then addition and subtraction.

Do you always work from left to right?

Only for operations of equal rank. Multiply and divide before you add and subtract.

What is 9 + 2 × 7?

\(9 + 14 = 23\).

Which comes first, addition or subtraction?

Neither: work them from left to right.

Why do we need these rules?

So that everyone gets the same answer from the same number sentence.

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