Year 6 Maths Number Sentences & Order of Operations

Order of operations & grouping symbols - Multiple operations

20 practice questions 0 video lessons Theory + worked examples

Every question with a fully worked solution.

Theory

Longer number sentences use the same rules: brackets first, then $\times$ and $\div$ from left to right, then $+$ and $-$ from left to right. Working one step at a time and rewriting the line each time keeps the work clear. This Year 6 maths page shows how to evaluate number sentences with several operations and how to write one number sentence for a multi-step problem.

To evaluate a number sentence means to work out its value.

The order: 1. brackets; 2. multiplication and division, left to right; 3. addition and subtraction, left to right.

Multiplication and division are equal in rank, so in $20 \div 4 \times 3$ the division is done first because it comes first: $5 \times 3 = 15$.

A multi-step word problem can be written as one number sentence using brackets, such as $50 - (5 \times 4 + 3 \times 3)$.

One step per line: $3 \times (8 + 2) - 14 \div 7 = 28$.

Work through the boxes in order, left to right inside each box.

The full order of operations:

\[ (\;) \;\rightarrow\; \times, \div \;\rightarrow\; +, - \]

brackets \to \times \div \to + -

\[ \text{equal rank: left to right} \]

equal rank left to right

\[ 7 + 20 \div 4 \times 3 = 7 + 5 \times 3 = 7 + 15 = 22 \]

7 + 20 \div 4 \times 3 = 22

          Rewrite the line each step. Doing one operation per line stops numbers getting lost.
        

          Estimate to check. \(6 \times 7 - 20 \div 4\) is a bit less than \(42\); \(37\) makes sense.
        

Evaluating a long number sentence

Brackets: work out each pair (innermost first).
Multiply and divide from left to right.
Add and subtract from left to right.
Rewrite the line after each step.

Writing one number sentence for a word problem

Write each part as a small calculation (such as $5 \times 4$).
Join them with the right signs, and add brackets around any part that must be done first.

Example 1 — Three operations

Evaluate $6 + 4 \times 5 - 7$.

Solution

$6 + 4 \times 5 - 7$	$=$	$6 + 20 - 7$
$6 + 20 - 7$	$=$	$19$

6 + 20 - 7 = 19

$19$.

Example 2 — Brackets and division

Evaluate $3 \times (8 + 2) - 14 \div 7$.

Solution

$3 \times 10 - 14 \div 7$	$=$	$30 - 2$
$30 - 2$	$=$	$28$

30 - 2 = 28

$28$, as in the first diagram.

Example 3 — Divide then multiply

Evaluate $7 + 20 \div 4 \times 3$.

Solution

$7 + 20 \div 4 \times 3$	$=$	$7 + 5 \times 3$
$7 + 5 \times 3$	$=$	$7 + 15$
$7 + 15$	$=$	$22$

7 + 15 = 22

$22$. The division comes first because it is on the left.

Example 4 — Three to try

Evaluate each.
(i) $30 - 8 \div 2 + 6$
(ii) $6 \times 7 - 20 \div 4$
(iii) $16 \div 8 + 5 \times 3$

Solution

$30 - 4 + 6$	$=$	$32$
$42 - 5$	$=$	$37$
$2 + 15$	$=$	$17$

32, 37, 17

(i) $32$
(ii) $37$
(iii) $17$

Example 5 — Bracket then left to right

Evaluate $64 \div (5 + 3) \times 2$.

Solution

$64 \div 8 \times 2$	$=$	$8 \times 2$
$8 \times 2$	$=$	$16$

64 \div 8 \times 2 = 16

$16$.

Example 6 — Shopping

Pies cost $\$4$ and drinks cost $\$3$. Lin buys $5$ pies and $3$ drinks and pays with $\$50$.
(i) Write one number sentence for her change.
(ii) Work it out.

Solution

$50 - (5 \times 4 + 3 \times 3)$	$=$	$50 - 29$
$50 - 29$	$=$	$21$

50 - (20 + 9) = 21

(i) $50 - (5 \times 4 + 3 \times 3)$
(ii) $\$21$

Example 7 — Eggs

A farm has $5$ trays of $12$ eggs. $12$ eggs break. The rest are packed in boxes of $8$. How many boxes are filled?

Solution

$(5 \times 12 - 12) \div 8$	$=$	$48 \div 8$
$48 \div 8$	$=$	$6$

(60 - 12) \div 8 = 6

$6$ boxes.

Example 8 — Place the brackets

Place one pair of brackets in $6 + 10 \div 2 \times 4$ so that its value is $32$.

Solution

$(6 + 10) \div 2 \times 4$	$=$	$8 \times 4$
$8 \times 4$	$=$	$32$

(6 + 10) \div 2 \times 4 = 32

$(6 + 10) \div 2 \times 4 = 32$. Without brackets the value is $26$.

Example 9 — Find the mistake

Kai evaluated $5 + 2 \times 6 - 3$ from left to right and wrote $39$. What is right?

Solution

$5 + 2 \times 6 - 3$	$=$	$5 + 12 - 3$
$5 + 12 - 3$	$=$	$14$

5 + 12 - 3 = 14

He added before multiplying. The answer is $14$.

Common pitfalls

Multiplying before dividing every time. $\times$ and $\div$ are equal: go left to right. $20 \div 4 \times 3 = 15$; doing $4 \times 3$ first is wrong.

Adding before multiplying. $5 + 2 \times 6 - 3 = 14$, not $39$.

Losing a number. Rewrite the whole line after every step.

Missing brackets in a word problem. Change from $\$50$ needs $50 - (20 + 9)$, not $50 - 20 + 9$.

Frequently asked questions

What order do I use for long number sentences?

Brackets, then × and ÷ left to right, then + and − left to right.

Is multiplication always before division?

No. They are equal; do whichever comes first from the left.

What is 7 + 20 ÷ 4 × 3?

$7 + 5 \times 3 = 7 + 15 = 22$.

How do I write one number sentence for a problem?

Write each part as a small calculation and join them, using brackets for parts that must be done first.

How can I check my answer?

Estimate first, and redo the steps one line at a time.

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More in Number Sentences & Order of Operations

\(6 + 4 \times 5 - 7\)	\(=\)	\(6 + 20 - 7\)
\(6 + 20 - 7\)	\(=\)	\(19\)

\(3 \times 10 - 14 \div 7\)	\(=\)	\(30 - 2\)
\(30 - 2\)	\(=\)	\(28\)

\(7 + 20 \div 4 \times 3\)	\(=\)	\(7 + 5 \times 3\)
\(7 + 5 \times 3\)	\(=\)	\(7 + 15\)
\(7 + 15\)	\(=\)	\(22\)

\(30 - 4 + 6\)	\(=\)	\(32\)
\(42 - 5\)	\(=\)	\(37\)
\(2 + 15\)	\(=\)	\(17\)

\(64 \div 8 \times 2\)	\(=\)	\(8 \times 2\)
\(8 \times 2\)	\(=\)	\(16\)

\(50 - (5 \times 4 + 3 \times 3)\)	\(=\)	\(50 - 29\)
\(50 - 29\)	\(=\)	\(21\)

\((5 \times 12 - 12) \div 8\)	\(=\)	\(48 \div 8\)
\(48 \div 8\)	\(=\)	\(6\)

\((6 + 10) \div 2 \times 4\)	\(=\)	\(8 \times 4\)
\(8 \times 4\)	\(=\)	\(32\)

\(5 + 2 \times 6 - 3\)	\(=\)	\(5 + 12 - 3\)
\(5 + 12 - 3\)	\(=\)	\(14\)

Order of operations & grouping symbols - Multiple operations

📖 Theory

Evaluating a long number sentence

Writing one number sentence for a word problem

Common pitfalls

Frequently asked questions

Theory