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Order of operations

Why 3 + 4 × 2 is 11 and not 14: brackets first, then powers, then multiplying and dividing, then adding and subtracting.

What is 3 + 4 × 2? If you work from left to right, you get 7 × 2 = 14. If you multiply first, you get 3 + 8 = 11. Both cannot be right. So mathematicians all over the world agree on one order for the operations, which is the name for adding, subtracting, multiplying, dividing and powers. With this order, everyone who reads a calculation gets the same answer. The answer here is 11. Scientific calculators and spreadsheets follow the same order too. (A very simple calculator may just work from left to right, so be careful with one.)

The order

Remember

Brackets, then powers, then multiply and divide, then add and subtract.

Multiply and divide are equal partners: do them from left to right. Add and subtract are equal partners too.

StepWhatLooks like
1Brackets(8 + 4)
2Powers3²
3Multiply and divide, left to right4 × 2, 36 ÷ 4
4Add and subtract, left to right3 + 8, 36 − 6

Many schools teach a word to remember this order. In Britain it is BODMAS or BIDMAS: Brackets, Orders (or Indices, other names for powers), Division, Multiplication, Addition, Subtraction. In America it is PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. They all say the same thing.

Multiply before you add

The most common question has only two operations: an adding and a multiplying. Do the multiplying first, then the adding.

Example 1

What is 10 + 2 × 5?

  1. Multiply first: 2 × 5 = 10.
  2. Then add: 10 + 10 = 20.

Answer: 20

Brackets come first

Brackets are the curved signs ( and ). They mean "do this part first". They are how you tell the reader that you want a different order. Without brackets, 8 + 4 × 3 is 20. With brackets, (8 + 4) × 3 is 12 × 3 = 36.

Example 2

What is (8 + 4) × 3 − 6?

  1. Brackets first: 8 + 4 = 12. Now the line reads 12 × 3 − 6.
  2. Multiply: 12 × 3 = 36. Now it reads 36 − 6.
  3. Subtract: 36 − 6 = 30.

Answer: 30

Example 3

What is 36 ÷ 4 + 2 × 5?

  1. There are no brackets or powers.
  2. Divide and multiply: 36 ÷ 4 = 9 and 2 × 5 = 10. Now it reads 9 + 10.
  3. Add: 9 + 10 = 19.

Answer: 19

Powers come before multiplying

A power is written with a small raised number. 3² means "3 multiplied by itself", so 3² = 3 × 3 = 9. It is called "3 squared". Powers are done after brackets, but before multiplying, dividing, adding and subtracting.

Example 4

What is 2 + 3² × 2?

  1. Power first: 3² = 9. Now it reads 2 + 9 × 2.
  2. Multiply: 9 × 2 = 18. Now it reads 2 + 18.
  3. Add: 2 + 18 = 20.

Answer: 20

Example 5

What is 3 × (4 + 2)² ÷ 9?

  1. Brackets first: 4 + 2 = 6. Now it reads 3 × 6² ÷ 9.
  2. Power: 6² = 36. Now it reads 3 × 36 ÷ 9.
  3. Multiply and divide from left to right: 3 × 36 = 108, then 108 ÷ 9 = 12.

Answer: 12

When the brackets have a power on them, the whole bracket is squared. (2 + 3)² is 5² = 25. That is not the same as 2² + 3² = 4 + 9 = 13.

Example 6

What is (2 + 3)² − 4²?

  1. Brackets first: 2 + 3 = 5. Now it reads 5² − 4².
  2. Powers: 5² = 25 and 4² = 16. Now it reads 25 − 16.
  3. Subtract: 25 − 16 = 9.

Answer: 9

Try it yourself

  1. 1.What is 8 + 6 × 3?

    Show the answer

    26

    Multiply first: 6 × 3 = 18. Then add: 8 + 18 = 26.

  2. 2.What is 24 ÷ 4 × 2?

    Show the answer

    12

    Divide and multiply have the same rank, so go from left to right: 24 ÷ 4 = 6, then 6 × 2 = 12.

  3. 3.What is (7 + 5) ÷ 3 + 4?

    Show the answer

    8

    Brackets first: 7 + 5 = 12. Then 12 ÷ 3 = 4, and 4 + 4 = 8.

  4. 4.What is 50 − 2 × 4²?

    Show the answer

    18

    Power first: 4² = 16. Then 2 × 16 = 32, and 50 − 32 = 18.

  5. 5.What is (9 − 3)² ÷ 4 + 1?

    Show the answer

    10

    Brackets: 9 − 3 = 6. Power: 6² = 36. Divide: 36 ÷ 4 = 9. Add: 9 + 1 = 10.

Ready to test yourself?

Play Arithmetic: 20 seconds per question, and it gets harder as you go.