Order of Operations

What you will learn

• Use the order brackets, powers, multiplication and division, then addition and subtraction.
• Work left to right when operations have equal rank.
• Use brackets to change the order.

Before you start: you should know multiplication, division, addition and subtraction. See Multiplication: Equal Groups and Arrays.

The idea

What is $2 + 3 \times 4$? If you add first you get $5 \times 4 = 20$. If you multiply first you get $2 + 12 = 14$. One expression cannot have two answers, so mathematicians agreed on an order.

The order

1. Brackets first.
2. Then powers, such as $3^2$.
3. Then multiplication and division, working left to right.
4. Then addition and subtraction, working left to right.

Multiplying and dividing have equal rank, and so do adding and subtracting. When the same rank turns up more than once, go from left to right.

Why multiplication goes first

Multiplication is repeated addition. $2 + 3 \times 4$ means $2$ plus three fours: $2 + 4 + 4 + 4 = 14$. That is why $3 \times 4$ is one unit.

Brackets change the order

Brackets say “do me first”. They are the way to ask for a different order.

Left to right

For $10 - 4 + 3$, subtract and add have equal rank, so go left to right: $10 - 4 = 6$, then $6 + 3 = 9$. Doing $4 + 3$ first would wrongly give $3$.

Worked examples

Example 1: Mixed operations

Work out $18 - 2 \times 5$.

1. Multiply first: $2 \times 5 = 10$.
2. Subtract: $18 - 10 = 8$.

Answer: $8$.

Example 2: With brackets and a power

Work out $3 \times (4 + 1)^2$.

1. Brackets: $4 + 1 = 5$.
2. Power: $5^2 = 25$.
3. Multiply: $3 \times 25 = 75$.

Answer: $75$.

Example 3: Left to right

Work out $36 \div 6 \times 2 + 1$.

1. Divide and multiply, left to right: $36 \div 6 = 6$, then $6 \times 2 = 12$.
2. Add: $12 + 1 = 13$.

Answer: $13$.

Do not always do the operations in the order they are written. Check for brackets, powers, multiplying and dividing first.

Why it matters

Calculators, spreadsheets and computer programs all use this order. Knowing it means that formulas such as $2\pi r + 5$ or $3x + 4$ mean the same to everyone.