Teach Me Maths

Order of Operations

On this page

How to work out a calculation in the right order: brackets, then powers and roots, then multiplying and dividing, then adding and subtracting, with equal levels worked from left to right. Also how to write brackets in a calculation and use a calculator safely.

The order of operations

When a calculation has more than one operation, everyone works in the same order, so that everyone gets the same answer.

  • First work out anything in brackets.
  • Then work out powers and roots, such as 3^2 or \sqrt{25}.
  • Then multiply and divide, working from left to right.
  • Last of all, add and subtract, working from left to right.

You may know this as BIDMAS or BODMAS. The I stands for indices, another name for powers. Multiplying and dividing are on the same level, and so are adding and subtracting.

In the exam you are usually asked to work out a calculation such as 2+5\times3, for 1 mark. You may also be asked to write brackets in a calculation to make it correct, or to explain what is wrong with someone's working.

Worked example 1: Multiply before adding

Work out 6+4\times3

  1. Multiplying comes before adding, so work out 4\times3=12 first.
  2. Keep the 6+ as it is: 6+4\times3=6+12

  3. Your turn. Now work out 6+12

    6+12=18

  4. Answer: 18

Brackets come first

Brackets group numbers together, and the part inside is worked out first. Compare these two calculations:

6+4\times3=6+12=18

(6+4)\times3=10\times3=30

The numbers are the same, but the brackets change the answer.

When there are brackets inside brackets, start with the inside pair and work outwards. Square brackets [\ ] work just like round ones:

2\times[10-(3+4)]=2\times[10-7]=2\times3=6

Equal levels go from left to right

Multiplying does not always come before dividing. When a calculation has only multiplying and dividing, work from left to right.

The same goes for adding and subtracting. In 12-5+2, the subtraction is on the left, so do it first: 12-5=7, then 7+2=9

Worked example 2: Divide and multiply from left to right

Work out 40\div5\times2

  1. Dividing and multiplying are on the same level. The one on the left is 40\div5
  2. 40\div5=8, so the calculation becomes 8\times2

  3. Your turn. Work out 8\times2

    8\times2=16

  4. Multiplying first would give 40\div10=4, which is wrong: that is the answer to 40\div(5\times2)
  5. Answer: 16

Powers and roots

A power tells you how many times a number is multiplied by itself. 4^2=4\times4=16, not 4\times2. Also 2^3=2\times2\times2=8

A square root undoes squaring. \sqrt{49}=7, because 7^2=49

A power belongs only to the number it is attached to. In 2\times3^2, only the 3 is squared: 2\times9=18. To square the whole of 2\times3 you would need brackets, (2\times3)^2=36

Put a negative number in brackets before you square it: (-3)^2=(-3)\times(-3)=9. Without brackets, -3^2 means square the 3 and then put the minus sign in front, which gives -9

Worked example 3: A power, a root and a multiplication

Work out 3\times4^2-\sqrt{25}

  1. Powers and roots first: 4^2=16 and \sqrt{25}=5
  2. The calculation is now 3\times16-5

  3. Your turn. Multiplying comes next. Work out 3\times16

    3\times16=48

  4. Last, the subtraction: 48-5=43
  5. Answer: 43

Fraction bars and calculators

A fraction bar groups the whole top and the whole bottom, like two pairs of brackets. In \frac{8+4}{7-3}, work out 8+4=12 and 7-3=4 before you divide: 12\div4=3

A calculator uses the order of operations too, so it only works out what you type. To work out \frac{18.6+5.4}{1.6\times2.5}, work out the top, 24, and the bottom, 4, and write them both down. Then 24\div4=6

Typing 24\div1.6\times2.5 gives 37.5, which is wrong: the calculator divides by 1.6 and then multiplies by 2.5. Use brackets, 24\div(1.6\times2.5), or the fraction button.

Worked example 4: Work out the top and bottom of a fraction

Work out \frac{7\times4-3}{2+3}

  1. Start with the top. Multiplying comes before subtracting: 7\times4=28

  2. Your turn. Work out the whole top, 28-3

    The top is 28-3=25

  3. The bottom is 2+3=5
  4. Now divide: 25\div5=5
  5. Answer: 5

Worked example 5: Write brackets to make a calculation correct

Write one pair of brackets in 5+3\times4-2=30 to make it correct.

  1. Your turn. First work out the calculation without brackets, 5+3\times4-2

    Without brackets: 5+3\times4-2=5+12-2=15

  2. 15 is too small. Adding before multiplying would make the answer bigger, so try brackets round 5+3
  3. (5+3)\times4-2=8\times4-2=32-2=30
  4. Answer: (5+3)\times4-2=30

Worked example 6: Use brackets in a problem

Five friends each buy a cinema ticket for £7 and a drink for £3. They pay with £60. Work out how much change they get.

  1. Each friend spends 7+3 pounds, and there are 5 friends, so the total cost is 5\times(7+3)
  2. Without the brackets, 5\times7+3=38 would pay for five tickets but only one drink.

  3. Your turn. Work out the total cost, 5\times(7+3), in pounds.

    5\times10=50, so the total cost is £50.

  4. The change is 60-50=10 pounds.
  5. Answer: £10

Common mistakes

  • Working from left to right through everything. In 4+5\times2, multiply first: the answer is 14, not 18.
  • Treating a power as multiplying. 3^4 means 3\times3\times3\times3=81, not 3\times4.
  • Squaring more than the power belongs to. In 2\times3^2, only the 3 is squared, so the answer is 18, not 36.
  • Leaving out brackets round a negative number. Write (-3)^2 when you square minus three, on paper and on a calculator.
  • Writing more than one pair of brackets. When the question asks for one pair, write just one pair, then check the answer.
  • Typing a fraction into a calculator in one go. Work out the top and the bottom separately and write them down, or use brackets.

Practice questions

More on this topic: Order of Operations worksheet with full solutions