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Learn Index · Arithmetic and Operations Grade 6–8

Mixed operations

Problems that combine addition, subtraction, multiplication and division.

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Putting it all together

Real problems rarely use just one operation. Mixed operations combine +, −, × and ÷ (and often powers, brackets, negative integers, decimals and fractions) in a single expression. The key is following the order of operations carefully.

Strategy

  1. Scan the whole expression first.
  2. Underline what must be done first (brackets, then powers).
  3. Do × and ÷ left to right.
  4. Do + and − left to right.
  5. Write one step per line.
  6. Check signs and estimate.

Example with integers

Evaluate: −4 × (3 − 8) + 12 ÷ (−3)

  1. Brackets: 3 − 8 = −5 → −4 × (−5) + 12 ÷ (−3)
  2. × and ÷: −4 × (−5) = 20; 12 ÷ (−3) = −4 → 20 + (−4)
  3. Add: 16

Example with decimals

Evaluate: 2.5 × 4 − 1.2 ÷ 0.3

  1. 2.5 × 4 = 10
  2. 1.2 ÷ 0.3 = 4
  3. 10 − 4 = 6

Example with fractions

Evaluate: 1/2 + 3/4 × 2/3

  1. Multiply first: 3/4 × 2/3 = 6/12 = 1/2
  2. Add: 1/2 + 1/2 = 1

Example with powers

Evaluate: 3 × 2³ − (10 − 4)² ÷ 4

  1. Brackets: 10 − 4 = 6
  2. Powers: 2³ = 8; 6² = 36
  3. × and ÷: 3 × 8 = 24; 36 ÷ 4 = 9
  4. Subtract: 24 − 9 = 15

Writing expressions from words

Words Expression
Add 5 to the product of 3 and 4 3 × 4 + 5
Multiply the sum of 3 and 4 by 5 (3 + 4) × 5
Subtract 6 from 20, then divide by 7 (20 − 6) ÷ 7

Brackets show what to do first when the words say "the sum of … multiplied by".

Real-life example

You buy 3 notebooks at $2.50 each and 2 pens at $1.25 each, and pay with a $20 bill. Change?

20 − (3 × 2.50 + 2 × 1.25) = 20 − (7.50 + 2.50) = 20 − 10 = $10

Worked examples

Example 1: 7 + 3 × (−2)

3 × (−2) = −6 → 7 − 6 = 1.

Example 2: (15 − 3) ÷ 4 + 2 × 5

12 ÷ 4 = 3; 2 × 5 = 10 → 13.

Example 3: 0.5 × 8 + 6 ÷ 0.5

4 + 12 = 16.

Common mistakes

  • Losing a negative sign partway through.
  • Adding before multiplying because the addition comes first in the line.
  • Leaving out brackets when translating words.

Try it yourself

  1. 9 − 2 × 3 + 4
  2. (−6) × 2 − 18 ÷ (−3)
  3. 2/3 × 9 − 1/2
  4. 4² − 3 × (5 − 7)
  5. Write and evaluate: "multiply the difference of 9 and 4 by 6".

Answers: 1) 7 2) −6 3) 5 1/2 4) 22 5) (9 − 4) × 6 = 30

Quick recap

  • Mixed operations follow the order of operations.
  • Write one step per line and track signs.
  • Use brackets when turning words into an expression.