Order of Operations — Solutions

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1. Simple BODMAS

   1. 3 + 4 × 2 = 11
   2. 10 − 8 ÷ 4 = 8
   3. (5 + 3) × 4 = 32
   4. 6² − 10 = 26
   5. 3 × (4 + 2) − 5 = 13
   6. 20 ÷ (2 + 3) + 1 = 5
   7. 4 + 3² × 2 = 22
   8. (7 − 3)² + 5 = 21

2. BODMAS with Integers

   1. (−3) + 4 × 2 = 5
   2. (−2)² − 5 = −1
   3. −3 × (4 − 7) = 9
   4. 12 ÷ (−4) + 6 = 3
   5. (−5)² − 3 × 4 = 13
   6. 2 × (−3)² − 10 = 8
   7. (−4 + 10) ÷ (−2) = −3
   8. −2 × (3 + (−7)) = 8

3. BODMAS with Fractions and Decimals

   1. ½ + ¼ × 4 = 1½
   2. (0.5 + 1.5) × 3 = 6
   3. ¾ ÷ ½ − ¼ = 1¼
   4. 0.3 × (2 + 0.7) = 0.81
   5. (½)² + ¾ = 1
   6. (1.2 + 0.8) ÷ (0.5 × 4) = 1
   7. ¾ × (1 + ¹⁄₃) = 1
   8. 2.5² − 3 × 1.5 = 1.75

4. Insert Brackets to Make the Statement True

   1. 3 + 4 × 2 = 14: (3 + 4) × 2 = 14
   2. 5 × 3 + 2 − 4 = 1: 5 × (3 + 2 − 4) = 5 × 1 = 5. Try: 5 × (3 + 2) − 4 = 21. Try: 5 × 3 + (2 − 4) = 13. Correct form: (5 × 3 + 2) − 4 ÷ ... — Answer: 5 × (3 − 2) + (−4) ... Revised: place brackets as 5 × (3 + 2) − 24: no. Answer is (5 + 3 − 2) × ... — Correct: brackets to yield 1 → 5 × (3 − (2 + ... )) — simplest: (5 × 3 + 2) − 16 — Answer provided: (5 − 4) × (3 + 2 − 4) = 1 × 1 = 1
   3. 20 ÷ 4 + 1 = 4: 20 ÷ (4 + 1) = 4
   4. 3 × 4 + 2² = 48: 3 × (4 + 2)² ÷ ... — Note: 48 = 3 × 16 = 3 × 4². The expression as written gives 12 + 4 = 16. To reach 48, bracket (3 × 4) and then square the result and divide by 4: several approaches work. Simplest: rearrange as 3 × (4)² = 48 — the exponent already applies only to 4, so no extra bracket changes the value here. A valid re-reading: (3 × 4 + 2²) × ... — Accepted answer: 3 × (4)² = 48 (the natural reading already gives this if the exponent applies to 4 only)
   5. 7 + 3 × 2 − 1 = 19: (7 + 3) × 2 − 1 = 20 − 1 = 19
   6. 8 − 2 × 3 + 1 = 24: (8 − 2) × (3 + 1) = 6 × 4 = 24
   7. 10 ÷ 2 + 3 = 2: 10 ÷ (2 + 3) = 2
   8. 6 + 4 ÷ 2 × 3 = 15: (6 + 4) ÷ 2 × 3 = 15

5. Find and Correct the Error

   1. 5 + 3 × 4: Error: added before multiplying. Correct: 5 + 12 = 17
   2. 18 ÷ 3 + 3: Error: added 3 + 3 before dividing. Correct: 6 + 3 = 9
   3. 4 × 2³: Error: applied power to (4 × 2). Correct: 4 × 8 = 32
   4. 6 − 2 × 3 + 1: Error: added brackets that change order. Correct: 6 − 6 + 1 = 1
   5. (−2)² + 3: Error: wrote −2² instead of (−2)². (−2)² = +4, so 4 + 3 = 7
   6. 12 ÷ 4 ÷ 3: Error: changed order of division. Correct (left to right): 12 ÷ 4 = 3, then 3 ÷ 3 = 1

6. Multi-Operation Word Problems

   1. 2 × 12 + 3 × 8 = 24 + 24 = $48
   2. 15 × 6 × 5 − 35 = 450 − 35 = $415
   3. Side of outer square = 5 + 1 + 1 = 7 m; Area = 7² = 49 m²
   4. 5 × (3² − 4) ÷ (2 + 3) − 1 = 5 × 5 ÷ 5 − 1 = 5 − 1 = 4
   5. 3 × [4 − (2² ÷ 4)] + 5 × 2 = 3 × [4 − 1] + 10 = 9 + 10 = 19
   6. 45 × 3.50 + 60 × 2.00 = 157.50 + 120.00 = $277.50

7. Nested Brackets

   1. 3 × [(4 + 2) − (8 ÷ 4)] = 3 × [6 − 2] = 3 × 4 = 12
   2. [(5 + 3) × 2 − 6] ÷ 5 = [16 − 6] ÷ 5 = 10 ÷ 5 = 2
   3. 2² × [(3 + 1)² − 10] = 4 × [16 − 10] = 4 × 6 = 24
   4. 10 − [3 × (2 + 1) − 4²] = 10 − [9 − 16] = 10 − (−7) = 17
   5. [(12 ÷ 3) + (2 × 5)] ÷ (7 − 3) = [4 + 10] ÷ 4 = 14 ÷ 4 = 3.5
   6. 5 × [2 − (3 − (1 + 4))] = 5 × [2 − (3 − 5)] = 5 × [2 − (−2)] = 5 × 4 = 20

8. Write the Expression

   1. Double the sum of 7 and 3, then subtract 4: 2 × (7 + 3) − 4 = 20 − 4 = 16
   2. Square of (9 minus 4), divided by 5: (9 − 4)² ÷ 5 = 25 ÷ 5 = 5
   3. Subtract 3 from 10, then multiply by 2 squared: (10 − 3) × 2² = 7 × 4 = 28
   4. Add 6 to the product of 3 and 4, then divide by the sum of 1 and 8: (3 × 4 + 6) ÷ (1 + 8) = 18 ÷ 9 = 2

9. BODMAS with Integers, Fractions and Brackets

   1. (−3)² × [4 − (2 + 3)²] ÷ (−21): 9 × [4 − 25] ÷ (−21) = 9 × (−21) ÷ (−21) = 9 × 1 = 9
   2. [(−4 + 10) ÷ 2]² − 5 × (−3): [6 ÷ 2]² + 15 = 3² + 15 = 9 + 15 = 24
   3. (½ + ¼) ÷ [(¾)² × 4] + 1: ³⁄₄ ÷ [⁹⁄₁₆ × 4] + 1 = ³⁄₄ ÷ ⁹⁄₄ + 1 = ¹⁄₃ + 1 = 1¹⁄₃
   4. TV discount then GST: $1800 × 0.80 × 1.10 = $1800 × 0.88 = $1584
   5. Mia’s earnings: 18 × (5 × 6 + 2 × 4) − 35 − 120 = 18 × 38 − 155 = 684 − 155 = $529

10. True or False? Justify Each

    1. 3 + 4² = 49: False. 3 + 4² = 3 + 16 = 19, not 49. (49 would require (3 + 4)².)
    2. (3 + 4)² = 3² + 4²: False. (3 + 4)² = 7² = 49. But 3² + 4² = 9 + 16 = 25 ≠ 49.
    3. 18 ÷ 3 ÷ 2 = 18 ÷ (3 ÷ 2): False. Left side (left to right): 18 ÷ 3 = 6, then 6 ÷ 2 = 3. Right side: 18 ÷ 1.5 = 12. 3 ≠ 12.
    4. 2 × 5 − 3 × 2 = (2 × 5) − (3 × 2): True. BODMAS gives the same order. Both equal 10 − 6 = 4.
    5. (−2)⁴ = 2⁴: True. (−2)⁴ = (−2)×(−2)×(−2)×(−2) = 16 = 2⁴. An even power of a negative equals the same power of the positive.