Order of operations (BODMAS)

If everyone evaluated $5 + 3 \times 2$ in their own order, the world would be in chaos. Luckily, mathematicians agreed long ago on a single priority list , captured by the friendly word BODMAS.

Idea

BODMAS stands for:

• Brackets , do the contents of brackets first.
• Of , handle "of" (a fancy word for multiplication, e.g., "half of $20$" $= 10$).
• Division and Multiplication , do these next, left to right, as they appear.
• Addition and Subtraction , do these last, left to right.

A key subtlety: D and M sit at the same level and are evaluated left to right. Similarly A and S are equal and left-to-right. So $20 \div 4 \times 2$ is $5 \times 2 = 10$ (not $20 \div 8 = 2.5$).

For $5 + 3 \times 2$: by BODMAS, multiplication first , $3 \times 2 = 6$ , then addition , $5 + 6 = 11$.

For nested brackets, work from the innermost outwards. $2 \times (3 + (4 \times 2)) = 2 \times (3 + 8) = 2 \times 11 = 22$.

You will encounter different bracket shapes: round $()$, curly $\{\}$, square $[]$. They all do the same job , group what is inside. By convention, when nested we write innermost as $()$, then $\{\}$, then $[]$, but any consistent use is fine.

The single best habit: write one step per line. Each line solves one operation. This makes mistakes easy to spot.

Worked examples

Example 1. Evaluate $5 + 3 \times 2 - 4 \div 2$.

By BODMAS, do $\times$ and $\div$ first: $5 + 6 - 2 = 9.$

Example 2. Evaluate $36 \div 6 \times 2$.

$\div$ and $\times$ are equal; go left to right. $36 \div 6 = 6, \quad 6 \times 2 = 12.$ Answer: $12$.

Example 3. Evaluate $20 - [2 + \{4 \times (3 + 1)\}]$.

Innermost bracket: $3 + 1 = 4$. Now $\{4 \times 4\} = 16$. Then $[2 + 16] = 18$. Finally $20 - 18 = 2$.

Example 4. Evaluate $\frac{1}{2} \text{ of } 36 + 5$.

"Of" is multiplication: $\frac{1}{2} \times 36 = 18$. Then $18 + 5 = 23$.

Try it yourself

1. Evaluate $7 + 2 \times 6$.
2. Evaluate $40 \div 8 + 3 \times 2$.
3. Evaluate $(8 + 4) \div 3$.
4. Evaluate $50 - [4 \times (3 + 2)]$.
5. Evaluate $3 + 4 \times 5 - 6 \div 2$.
6. Evaluate $\frac{3}{4}$ of $20 + 7$.
7. Evaluate $\{(15 - 3) \div 4\} \times 5$.
8. Put brackets in $8 - 4 \times 2$ to make the value $8$.

Activity

Take a calculator. Type $5 + 3 \times 2$ and press equals. Then type $(5 + 3) \times 2$. Compare the two answers , the calculator follows BODMAS automatically.

In pairs, one person writes a long expression without brackets; the other inserts brackets to get a different value. Compare results.