Solving questions using BODMAS

This lesson explains the BODMAS rule, which tells us the order in which to simplify a numerical expression, and then works through several examples that combine brackets, powers, division, multiplication, addition, and subtraction.

The BODMAS rule

When an expression mixes several operations, we do not simply work from left to right. Instead we follow the order set by the word BODMAS:

• B brackets
• O of (orders: powers and roots)
• D division
• M multiplication
• A addition
• S subtraction

We simplify in that order, starting with whatever is inside brackets.

Example 1: no brackets

Simplify $6 + 2 \times 7$.

There are no brackets, so we do the multiplication first, then the addition.

$6 + 2 \times 7 = 6 + 14 = 20$

Example 2: a bracket, then multiplication

Simplify $3 \times (2 + 4) - 8 \div 4$.

First the bracket, then division and multiplication, then subtraction.

$3 \times (2 + 4) - 8 \div 4 = 3 \times 6 - 2 = 18 - 2 = 16$

Example 3: division and multiplication before adding

Simplify $105 \div 5 - 2 + 3 \times 5$.

Do the division and the multiplication first.

$105 \div 5 - 2 + 3 \times 5 = 21 - 2 + 15 = 34$

Example 4: a longer mix

Simplify $72 + 10 \times 2 - 8 \div 2$.

Multiplication and division come before addition and subtraction.

$72 + 10 \times 2 - 8 \div 2 = 72 + 20 - 4 = 88$

Example 5: a bracket that is itself an expression

Simplify $15 \times (32 \div 2 \times 10) + 15$.

The bracket is its own expression, so simplify inside it first: division before multiplication.

$32 \div 2 \times 10 = 16 \times 10 = 160$

Then the multiplication and addition outside:

$15 \times 160 + 15 = 2400 + 15 = 2415$

Example 6: an expression with a power

Simplify $5 + 2 \times 3^2 \div 3 + 7 - 5$.

The order is the power first, then division and multiplication, then addition and subtraction.

$5 + 2 \times 3^2 \div 3 + 7 - 5 = 5 + 2 \times 9 \div 3 + 7 - 5$

$= 5 + 2 \times 3 + 7 - 5 = 5 + 6 + 7 - 5 = 13$

Example 7: nested square brackets

Simplify $10 \div 2 \times \big[(5 - 3) \div 2 \times (1 + 8)\big]$.

Work the simple brackets inside the square bracket first.

$(5 - 3) = 2, \qquad (1 + 8) = 9$

Then simplify inside the square bracket:

$2 \div 2 \times 9 = 1 \times 9 = 9$

Finally the outside:

$10 \div 2 \times 9 = 5 \times 9 = 45$

Example 8: a power with nested brackets

Simplify $10^2 + \big[(20 \div 5) \div 2 - (2 + 3 - 8)\big]$.

Start with the innermost brackets:

$20 \div 5 = 4, \qquad 2 + 3 - 8 = -3$

Then the square bracket:

$4 \div 2 - (-3) = 2 + 3 = 5$

Then the power and the addition:

$10^2 + 5 = 100 + 5 = 105$

Example 9: another nested bracket

The teacher closes with one more nested example. Following the spoken steps, the inner brackets combine to $10$, and the result is built up as $40 \times 10 = 400$.

Key takeaways

• BODMAS sets the order: brackets, then orders (powers and roots), then division and multiplication, then addition and subtraction.
• Always simplify what is inside brackets first, working from the innermost brackets outward.
• Reduce an expression one operation at a time, and the final value follows safely.