Order of Operations and Laws of Arithmetic

Why this matters

Without a calculator, the order you do a calculation in can turn it from painful to trivial. Knowing the rules of arithmetic — which steps you must do first and which you are free to rearrange — lets you spot the easy route.

What you are allowed to rearrange

Law	Example
Multiplication is commutative	$\frac{2}{3} \times 7.148 = 7.148 \times \frac{2}{3}$
Multiplication is associative	$(1.5 \times 3.5) \times 24 = 1.5 \times (3.5 \times 24)$
Distributive law	$a(b + c) = ab + ac$
Dividing is multiplying by the reciprocal	$\div 3 = \times \frac{1}{3}$

Brackets you do not have to open first

Wasim simplified $\left(5\frac{3}{17} - 2\frac{4}{11}\right) - \left(\frac{2}{17} - \frac{4}{11}\right)$ by removing the brackets and grouping the seventeenths and elevenths together:

$5\tfrac{3}{17} - \tfrac{2}{17} - 2\tfrac{4}{11} + \tfrac{4}{11} = 5\tfrac{1}{17} - 2 = 3\tfrac{1}{17}$

That is allowed, as long as you keep track of the signs — the minus in front of the second bracket changes $-\frac{4}{11}$ into $+\frac{4}{11}$ .

Remember. Addition and subtraction come last. In $1\frac{4}{5} + 2\frac{3}{4} \times 1\frac{1}{3}$ , do the multiplication first: $2\frac{3}{4} \times 1\frac{1}{3} = \frac{11}{3}$ , then add.

Key rules and formulas

BIDMAS still rules

Brackets, Indices, Division and Multiplication, Addition and Subtraction. A long fraction line acts as a bracket round the top and another round the bottom.

The distributive law

$a \times b + a \times c = a \times (b + c)$

2 \frac{3}{8} \times 7 + 2 \frac{3}{8} \times 5 = 2 \frac{3}{8} \times 12

— one multiplication instead of two.

Decimals as fractions

Writing

3.31 = \frac{331}{100}

1.5 = \frac{3}{2}

often lets numbers cancel.

1.5 \times 3.5 \times 24 = \frac{3}{2} \times \frac{7}{2} \times 24 = 3 \times 7 \times 6 = 126

Multiplication and division can be reordered

3 \times 7.148 \times \frac{2}{3} \div 2 = 7.148 \times (3 \times \frac{2}{3} \times \frac{1}{2}) = 7.148 \times 1 = 7.148

Worked examples

Worked example 1

A long fraction line

Calculate

\frac{( \frac{3}{4} ) ^{2}}{3 \frac{2}{5} - 1 \frac{3}{4}}

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

Using the distributive law

Work out

2 \frac{3}{8} \times 7 + 2 \frac{3}{8} \times 5

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

Decimals that cancel

Work out

3.31 \times 5^{2} + 9

Try it yourself first, then reveal the solution one step at a time.

Input	Output
4⅔	Row 1, Output
Row 2, Input	2⅕
10	Row 3, Output

Input

Output

4⅔

2⅕

Order of Operations and Laws of Arithmetic

In this lesson you will learn to…

Why this matters

What you are allowed to rearrange

Brackets you do not have to open first

Key rules and formulas

BIDMAS still rules

The distributive law

Decimals as fractions

Multiplication and division can be reordered

Worked examples

A long fraction line

Using the distributive law

Decimals that cancel

Practice: order of operations with fractions