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Arithmetic

The order of operations

In an expression with several operations, the order matters: 3+4×23 + 4 \times 2 is 1111, not 1414. Brackets first, then powers, then multiplication and division from left to right, and last addition and subtraction from left to right.

The rule, in four steps

  1. Brackets

    Work out everything inside brackets first.

  2. Powers

    Then powers, such as 323^2.

  3. Multiplication and division

    From left to right, in the order they appear.

  4. Addition and subtraction

    Last, also from left to right.

Multiplication does not always come before division, and addition does not always come before subtraction: within each pair, go from left to right.

Worked examples

Multiplication before addition

Problem: 3+4×23 + 4 \times 2

  1. Multiply first.

    3+83 + 8

  2. Then add.

    1111

Result: 1111

Brackets change the order

Problem: (3+4)×2(3 + 4) \times 2

  1. Brackets first.

    7×27 \times 2

  2. Then multiply.

    1414

Result: 1414

Left to right

Problem: 20−12÷4×220 - 12 \div 4 \times 2

  1. Division and multiplication, left to right: 12÷4=312 \div 4 = 3.

    20−3×220 - 3 \times 2

  2. Then 3×2=63 \times 2 = 6.

    20−620 - 6

  3. Subtract.

    1414

Result: 1414

Powers and brackets

Problem: 2×32−(8−3)2 \times 3^2 - (8 - 3)

  1. Brackets: 8−3=58 - 3 = 5.

    2×32−52 \times 3^2 - 5

  2. Power: 32=93^2 = 9.

    2×9−52 \times 9 - 5

  3. Multiply, then subtract.

    18−5=1318 - 5 = 13

Result: 1313

Common mistakes

  • Mistake 1 · Going strictly left to right: 3+4×2=7×2=143 + 4 \times 2 = 7 \times 2 = 14.

    Why it happens
    Multiplication comes before addition.
    What to do instead
    3+8=113 + 8 = 11.

  • Mistake 2 · Always multiplying before dividing: 12÷4×2=12÷812 \div 4 \times 2 = 12 \div 8.

    Why it happens
    Multiplication and division have the same priority, so they go left to right.
    What to do instead
    12÷4=312 \div 4 = 3, then 3×2=63 \times 2 = 6.

  • Mistake 3 · Applying the power to the product: 2×32=62=362 \times 3^2 = 6^2 = 36.

    Why it happens
    The power belongs only to the 33.
    What to do instead
    32=93^2 = 9, then 2×9=182 \times 9 = 18.

  • Mistake 4 · Adding before subtracting: 10−4+3=10−7=310 - 4 + 3 = 10 - 7 = 3.

    Why it happens
    Addition and subtraction have the same priority, so they go left to right.
    What to do instead
    10−4=610 - 4 = 6, then 6+3=96 + 3 = 9.

Practice

  1. 1. Calculate 6+18÷36 + 18 \div 3.

    Hint

    Divide first.

    Show answer

    6+6=126 + 6 = 12

  2. 2. Calculate (6+18)÷3(6 + 18) \div 3.

    Hint

    Brackets first.

    Show answer

    24÷3=824 \div 3 = 8

  3. 3. Calculate 5×22−45 \times 2^2 - 4.

    Hint

    The power belongs to the 22.

    Show answer

    5×4−4=165 \times 4 - 4 = 16

  4. 4. Calculate 30−6×2+430 - 6 \times 2 + 4.

    Hint

    Multiply, then go left to right.

    Show answer

    30−12+4=2230 - 12 + 4 = 22

Related concepts

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