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pemdasworksheet / Order of operations

Order of operations

The agreed order for simplifying an expression, so that everyone gets the same answer.

The rules

  1. PParentheses
  2. EExponents
  3. MDMultiply and divide, left to right
  4. ASAdd and subtract, left to right

Two rules carry most of the weight. First, multiplication and division are one level, and so are addition and subtraction: inside a level, go from left to right. Second, with nested grouping symbols, start with the innermost pair.

Why an order is needed

Read 2 + 3 × 4 from left to right and you get 20. Multiply first and you get 14. Both are reasonable; the convention picks 14, so a written expression means one thing to every reader. Grouping symbols let you override the order: (2 + 3) × 4 is 20.

Six worked examples

From grade 5 whole numbers to fractions and decimals. Each line does one operation.

  1. (14 + 20) − 2 = 32
    1. Parentheses: 14 + 20 = 3434 − 2
    2. Subtract: 34 − 2 = 3232
  2. [(17 − 12 + 53) × 5] ÷ 2 = 325
    1. Parentheses, exponents: 5³ = 125[(17 − 12 + 125) × 5] ÷ 2
    2. Parentheses, subtract (left to right): 17 − 12 = 5[(5 + 125) × 5] ÷ 2
    3. Parentheses: 5 + 125 = 130[130 × 5] ÷ 2
    4. Parentheses: 130 × 5 = 650650 ÷ 2
    5. Divide: 650 ÷ 2 = 325325
  3. −2 × [4 × (62 − 2)] = −272
    1. Parentheses, exponents: 6² = 36−2 × [4 × (36 − 2)]
    2. Parentheses: 36 − 2 = 34−2 × [4 × 34]
    3. Parentheses: 4 × 34 = 136−2 × 136
    4. Multiply: −2 × 136 = −272−272
  4. (11 + 10 − 196 ÷ (−7)2) × (19 + 5) = 408
    1. Parentheses, exponents: (−7)² = 49(11 + 10 − 196 ÷ 49) × (19 + 5)
    2. Parentheses, divide: 196 ÷ 49 = 4(11 + 10 − 4) × (19 + 5)
    3. Parentheses, add (left to right): 11 + 10 = 21(21 − 4) × (19 + 5)
    4. Parentheses: 21 − 4 = 1717 × (19 + 5)
    5. Parentheses: 19 + 5 = 2417 × 24
    6. Multiply: 17 × 24 = 408408
  5. (12 ÷ 6) × 34 + 14 = 516
    1. Parentheses: 1⁄2 ÷ 6 = 1⁄12112 × 34 + 14
    2. Multiply: 1⁄12 × 3⁄4 = 1⁄16116 + 14
    3. Add: 1⁄16 + 1⁄4 = 5⁄16516
  6. 8.02 + (8 − 6) = 10.02
    1. Parentheses: 8 − 6 = 28.02 + 2
    2. Add: 8.02 + 2 = 10.0210.02

Tips for checking work

  • Rewrite the whole expression after each step, as the answer key does. Most mistakes happen when two steps are done in one line.
  • Underline the operation you are about to do before doing it.
  • Check that the number of operations goes down by exactly one on each line.

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Questions

What is the order of operations?

It is the agreed order for simplifying an expression with more than one operation: grouping symbols first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. Without it, 2 + 3 × 4 could be read as 20 or 14; the agreed answer is 14.

In PEMDAS, does multiplication come before division?

No. Multiplication and division share one step and are done from left to right, whichever comes first. The same goes for addition and subtraction. So 24 ÷ 4 × 2 is 12, not 3. BODMAS lists D before M for the same reason PEMDAS lists M before D: a mnemonic needs some order.

Do PEMDAS, GEMDAS and BODMAS give different answers?

No. They are three names for the same rules. PEMDAS says parentheses, GEMDAS says grouping symbols, BODMAS says brackets and orders. Every problem on this site has one answer whichever name you pick; only the words in the instructions and the answer-key labels change.

Which grades is this for?

The presets follow what US classrooms usually cover: grade 5 uses whole numbers with parentheses, brackets and braces; grade 6 adds exponents; grade 7 adds negative numbers; grade 8 adds longer expressions and negative bases such as (−2)³. Fraction and decimal sheets work with any grade preset.

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