Order of Operations
Every question with a fully worked solution.
- Order of Operations - Video - Order Of Operations Watch
Theory
The order of operations is the agreed order for working out an expression, so that everyone gets the same answer. This Year 7 maths page explains BODMAS (brackets, orders, division and multiplication, addition and subtraction) and uses it with negative numbers, fraction bars, powers and missing numbers.
Use BODMAS to remember the order:
| Letter | Step |
|---|---|
| B | Brackets first (and other grouping symbols, such as a fraction bar) |
| O | Orders: powers and square roots |
| DM | Division and Multiplication, working from left to right |
| AS | Addition and Subtraction, working from left to right |
A fraction bar is a grouping symbol: work out the whole top and the whole bottom, then divide.
Use the integer rules at every step. Subtracting a negative is the same as adding: \(-(-4) = +4\).
The order, from first to last:
How to evaluate an expression
- Brackets and other grouping symbols first.
- Powers next.
- Multiply and divide from left to right.
- Add and subtract from left to right. Write one step per line.
How to find a missing number
- Simplify every part you can.
- Undo the operations in reverse order, doing the same to both sides.
- Check by putting the number back in.
Multiply and divide first, then add and subtract from left to right.
| \(7 \times (-2)\) | \(=\) | \(-14\) |
| \(4 \div 2\) | \(=\) | \(2\) |
| \(6 - 14 - 2\) | \(=\) | \(-8 - 2\) |
| \(\) | \(=\) | \(-10\) |
\(-10\)
(i) \(-4 \times (3 - 9)\)
(ii) \(3 - \dfrac{14 - 2}{1 - 5}\)
| \(-4 \times (3 - 9)\) | \(=\) | \(-4 \times (-6) = 24\) |
| \(3 - \dfrac{14 - 2}{1 - 5}\) | \(=\) | \(3 - \dfrac{12}{-4}\) |
| \(\) | \(=\) | \(3 - (-3) = 6\) |
(i) \(24\)
(ii) \(6\)
(i) \((-5)^2 - 30\)
(ii) \(-2^2 + 3 \times (-2)\)
| \((-5)^2 - 30\) | \(=\) | \(25 - 30\) |
| \(\) | \(=\) | \(-5\) |
| \(-2^2 + 3 \times (-2)\) | \(=\) | \(-4 + (-6)\) |
| \(\) | \(=\) | \(-10\) |
(i) \(-5\)
(ii) Only the \(2\) is squared: \(-10\)
| \(4 \times \square - 21 - 8\) | \(=\) | \(3\) |
| \(4 \times \square - 29\) | \(=\) | \(3\) |
| \(4 \times \square\) | \(=\) | \(32\) |
| \(\square\) | \(=\) | \(8\) |
Check: \(4 \times 8 - 21 - 8 = 3\). The missing number is \(8\).
Green squares: \(5 \times 3 = 15\). Then undo the operations.
| \(15 + \square \times (-2)\) | \(=\) | \(-7\) |
| \(\square \times (-2)\) | \(=\) | \(-22\) |
| \(\square\) | \(=\) | \(11\) |
Check: \(5 \times 3 + 11 \times (-2)\) \(= 15 - 22 = -7\). He landed on \(11\) red squares.
(i) The product of \(-3\) and the sum of \(5\) and \(-8\).
(ii) The sum of \(-3 \times 5\) and \(-8\).
| \(-3 \times (5 + (-8))\) | \(=\) | \(-3 \times (-3)\) |
| \(\) | \(=\) | \(9\) |
| \(-3 \times 5 + (-8)\) | \(=\) | \(-15 + (-8)\) |
| \(\) | \(=\) | \(-23\) |
(i) \(9\)
(ii) \(-23\)
Common pitfalls
Frequently asked questions
What does BODMAS stand for?
Brackets, Orders (powers and roots), Division and Multiplication, Addition and Subtraction.
Do you always multiply before dividing?
No. Multiplication and division have equal priority. Work from left to right.
Is a fraction bar like brackets?
Yes. Work out the top and the bottom first, then divide.
What is -3 squared?
Without brackets, \(-3^2 = -9\). With brackets, \((-3)^2 = 9\).
Why do we need an order of operations?
So that everyone works out the same expression in the same way and gets the same answer.