1.4 Priority of operations and inverse operations — Learn

Prerequisites: the four operations (section 1.2); negative numbers (1.3).

By the end of this section you can: evaluate an expression in the correct order; use brackets to change the order; and check a calculation with an inverse operation or by reasoning backwards.

1. The conventional order

The order is BIDMAS:

Brackets
Indices (powers and roots)
Division and Multiplication — equal priority, left to right
Addition and Subtraction — equal priority, left to right

Example 1. Work out $3 + 2 \times 4$ .

Multiplication first: $2 \times 4 = 8$ , then $3 + 8 = 11$ . The answer is $11$ , not $20$ .

Example 2. Work out $(3 + 2) \times 4$ .

Brackets first: $3 + 2 = 5$ , then $5 \times 4 = 20$ .

So brackets can change an answer completely: $3 + 2 \times 4 = 11$ , but $(3 + 2) \times 4 = 20$ .

Watch out. Thinking multiplication always beats division. They have equal priority, so work left to right: $12 \div 4 \times 3 = 3 \times 3 = 9$ , not $12 \div 12 = 1$ . Addition and subtraction behave the same way: $10 - 4 + 2 = 8$ .

2. Indices, roots and reciprocals

Example 3. Work out $2 + 3^2$ .

Index first: $3^2 = 9$ , then $2 + 9 = 11$ .

Example 4. Work out $\sqrt{16} + 2 \times 3$ .

Root first: $\sqrt{16} = 4$ ; multiplication next: $2 \times 3 = 6$ ; then $4 + 6 = 10$ .

3. Inverse operations

Each operation is undone by its inverse:

Operation	Inverse
$+$	$-$
$\times$	$\div$
squaring	square root

You can use inverses to check an answer, and to reason backwards to find a missing number.

Example 5. I think of a number, multiply it by 4, then add 7. The result is $31$ . What was the number?

Work backwards, undoing each step:

31 - 7 = 24 \qquad 24 \div 4 = 6

The number was $6$ . Check forwards: $6 \times 4 + 7 = 24 + 7 = 31$ . ✓

4. Progress check

Write the stall's profit as a single expression, then evaluate it in the correct order.

\underbrace{95 \times 1.20}_{\text{drinks}} + \underbrace{6 \times 3.00}_{\text{cakes}} - \underbrace{(120 \times 0.65 + 8 \times 1.85)}_{\text{stock}}

Brackets first for the stock, then the multiplications, then add and subtract:

114 + 18 - (78 + 14.80) = 132 - 92.80 = £39.20

You should now be able to

apply BIDMAS, including equal priority left to right;
use brackets to change the order of operations;
check an answer with an inverse operation;
work backwards to find a missing number.