Ordering Mathematical Operations | BODMAS

See also: Positive and Negative Numbers

With a simple sum that only has two numbers and one single operation, or sign, it’s easy to see how to calculate the answer. Either you add, subtract, multiply, or divide.

But what about when there are several numbers, and different operations? Maybe you need to divide and multiply, or add and divide. What do you do then?

Fortunately, mathematics is a logic-based discipline. As so often, there are some simple rules to follow that help you work out the order in which to do the sum.


Rules of Ordering in Mathematics - BODMAS

BODMAS - Rules of Ordering in Mathematics.  Brackets, Orders, Division, Multiplication, Addition and Subtraction.

BODMAS is a useful acronym that lets you know which order to solve mathematical problems (or sums). It's important that you follow the rules of BODMAS as without it your answers can be wrong.

The BODMAS acronym is for:

  • Brackets (parts of a calculation inside brackets always come first).
  • Orders (numbers involving powers or square roots).
  • Division.
  • Multiplication.
  • Addition.
  • Subtraction.

Brackets

Start with anything inside brackets, going from left to right.

Example:

4 × (3 + 2) = ?

You need to do the operation, or sum, inside the brackets first, 3 + 2, then multiply the answer by 4.

3 + 2 = 5.
4 × 5 = 20

If you ignored the brackets and did the sum 4 × 3 + 2 you would get 14. You can see how the brackets make a difference to the answer.

Orders

Do anything involving a power or a square root next (these are also known as orders), again working from left to right if there is more than one.

Example:

32 + 5 = ?

You need to do the power sum first, before you can add 5.

32 = 3 × 3 = 9
9 + 5 = 14

Division and Multiplication

Once you have done any parts of the sum involving brackets or powers the next step is division and multiplication.

Multiplication and division rank equally, so you go from left to right in the sum, doing each operation in the order in which it appears.

Example:

4 × 5 ÷ 2 + 7 = ?

You need to do division and multiplication first, but you have one of each.

Start from the left and work across to the right, which means that you start with 4 × 5 = 20. Then do the division, 20 ÷ 2 = 10.

Only then do you move to the addition: 10 + 7 = 17. The answer is 17.


See our pages: Multiplication and Division for more.

Addition and Subtraction

The final step is to calculate any addition or subtraction. Again, subtraction and addition rank equally, and you simply move from left to right.

Example:

4 + 6 - 7 + 3 = ?

You simply start on the left and work your way across.

4 + 6 = 10
10 - 7 = 3
3 + 3 = 6
The answer is 6.


See our pages: Addition and Subtraction for more.


Together, these four rules give you BODMAS:

Brackets,

Orders (powers and roots),

Division and Multiplication,

Addition and Subtraction.

When all your operations are the same level (for example, all division or multiplication), simply work from left to right.


PEDMAS

PEDMAS is commonly used the in USA it works the same as BODMAS. The PEDMAS acronym is:

Parentheses,

Exponents (powers and roots),

Division and Multiplication,

Addition and Subtraction.


BODMAS Test Questions

The rules of BODMAS are easiest to understand with some practice and examples.

Try these sums yourself and then open up the box (click on the + symbol to the left of the sums) to see the workings and answers.

There are no brackets or orders in this sum.

  1. Multiplication comes before addition, so you start with 20 × 3 = 60.
  2. Your sum now reads 3 + 60

The answer is therefore 63.

  1. Start with brackets. (3 + 2) = 5.
  2. Your sum now reads 25 - 5 ÷ 5
  3. Division comes before subtraction. 5 ÷ 5 = 1.
  4. Your sum now reads 25 - 1

The answer is therefore 24.

  1. Start with brackets. (1+10) = 11.
  2. Your sum now reads 10 + 6 × 11
  3. Multiplication comes before addition. 6 × 11 = 66.
  4. Your sum now reads 10 + 66.

The answer is therefore 76.

When there is no sign like in this sum, the operator is a multiplication so the sum is 5 × (3 + 2) + 52.

  1. You need to do the brackets first: (3 + 2) = 5.
  2. That gives you 5 × 5 + 52.
  3. The next step is orders, in this case, the square. 52 = 5 × 5 = 25. Now you have 5 × 5 + 25.
  4. Division and multiplication come before addition and subtraction, so your next step is 5 × 5 = 25. Your sum now reads 25 + 25 = 50.

The answer is 50.

This one has everything! But don’t panic. BODMAS still applies, and all you have to do is unpick the sum.

  1. Start with brackets. (105 + 206) = 311.
  2. Your sum now reads 311 – 550 ÷ 52 + 10
  3. Next, orders or powers. In this case, that’s 52 = 25.
  4. The sum now reads 311 – 550 ÷ 25 + 10
  5. Next, division and multiplication. There is no multiplication, but the division is 550 ÷ 25 = 22.
  6. Your sum now reads 311 – 22 + 10.
  7. Although you still have two operations left, addition and subtraction rank equally, so you just go from left to right. 311 - 22 = 289, and 289 + 10 = 299.

The answer is 299.

Sums like this often do the rounds on Facebook and other social media sites, with captions like '90% of people get this wrong'. Just follow the rules of BODMAS to get the correct answer.

  1. There are no brackets or orders so start with division and multiplication.
  2. 7 ÷ 7 = 1 and 7 × 7 = 49.
  3. The sum now reads 7 + 1 + 49 - 7
  4. Now do the addition and subtraction. 7 + 1 + 49 = 57 - 7 = 50

The answer is therefore 50.


How Did You Do?

Hopefully you managed to get all the answers right. If not, go back and review where you went wrong, and read over the rules again.

The more you practice, the easier BODMAS becomes and eventually you won’t even have to think about it any more.

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