Finding Percentages of Quantities — Solutions

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1. Basic % of quantities

   1. 50% of 80: 40
   2. 25% of 120: 30
   3. 10% of 350: 35
   4. 75% of 200: 150
   5. 20% of 60: 12
   6. 5% of 180: 9
   7. 30% of 90: 27
   8. 40% of 150: 60

2. Decimal multiplier method

   1. 35% of 140: 0.35 × 140 = 49
   2. 65% of 80: 0.65 × 80 = 52
   3. 12% of 250: 0.12 × 250 = 30
   4. 8% of 75: 0.08 × 75 = 6
   5. 45% of 360: 0.45 × 360 = 162
   6. 7.5% of 200: 0.075 × 200 = 15
   7. 3% of 400: 0.03 × 400 = 12
   8. 110% of 50: 1.10 × 50 = 55

3. Finding the whole (unitary method)

   1. 10% is 7: 1% = 0.7, so 100% = 70
   2. 25% is 18: 1% = 0.72, so 100% = 72
   3. 5% is 12: 1% = 2.4, so 100% = 240
   4. 40% is 56: 1% = 1.4, so 100% = 140
   5. 20% is 35: 1% = 1.75, so 100% = 175
   6. 15% is 24: 1% = 1.6, so 100% = 160

4. Mixed problems

   1. 75% of 32: 0.75 × 32 = 24 students
   2. 30% of $180: 0.30 × 180 = $54 discount
   3. 15% of $620: 0.15 × 620 = $93 saved per week
   4. 5% of 840: 0.05 × 840 = 42 students absent

5. Problem solving

   1. GST on $960: 10% of $960 = $96 GST. Total = $960 + $96 = $1056
   2. 78% of 50 marks: 0.78 × 50 = 39 marks
   3. Cream: 40% of 250 = 100 mL used. Remaining = 150 mL. Then 25% of 150 = 37.5 mL used. Remaining = 112.5 mL.
   4. 35% of 160 vs 40% of 140: 35% of 160 = 56. 40% of 140 = 56. They are equal!

6. Special percentages (>100% and <1%)

   1. 125% of 80: 1.25 × 80 = 100
   2. 200% of 45: 2.00 × 45 = 90
   3. 0.5% of 600: 0.005 × 600 = 3
   4. 150% of 120: 1.50 × 120 = 180
   5. 0.25% of 800: 0.0025 × 800 = 2
   6. 250% of 36: 2.50 × 36 = 90
   7. 175% of 200: 1.75 × 200 = 350
   8. 0.1% of 5000: 0.001 × 5000 = 5

7. Mental strategy practice

   1. 15% of 80: 10% = 8; 5% = 4; 15% = 12
   2. 35% of 200: 10% = 20; 30% = 60; 5% = 10; 35% = 70
   3. 45% of 40: 10% = 4; 40% = 16; 5% = 2; 45% = 18
   4. 5% of 360: 10% = 36; 5% = 18
   5. 22% of 50: 10% = 5; 20% = 10; 2% = 1; 22% = 11
   6. 12% of 150: 10% = 15; 2% = 3; 12% = 18

8. Unitary method extended

   1. 30% is 45: 1% = 1.5; 100% = 150
   2. 8% is $36: 1% = $4.50; 100% = $450
   3. 60% is 150 kg: 1% = 2.5 kg; 100% = 250 kg
   4. 75% is 24 students: 1% = 0.32; 100% = 32 students in the class
   5. 7% tax is $14: 1% = $2; 100% = $200 pre-tax price
   6. 12.5% is 25 km: 1% = 2 km; 100% = 200 km full journey

9. Compare and reason

   1. 20% of 250 vs 25% of 200: 20% of 250 = 50. 25% of 200 = 50. They are equal!
   2. Bicycle owners: School A: 60% of 400 = 240. School B: 45% of 550 = 247.5 ≈ 248. School B has more bicycle owners.
   3. Mark-up comparison: Item A: 30% of $40 = $12. Item B: 20% of $65 = $13. Item B has the larger dollar mark-up.
   4. Investment interest: Investment X: 5% of $1600 = $80. Investment Y: 8% of $1000 = $80. Both earn the same interest ($80).

10. Multi-step percentage problems

    1. Factory units exported: Rejected: 4% of 1200 = 48. Remaining: 1152. Exported: 15% of 1152 = 172.8 ≈ 173 units.
    2. Leo’s spending: Tax: 22% of $850 = $187. After tax: $663. Savings: 10% of $663 = $66.30. Spending: $663 − $66.30 = $596.70.
    3. Students cycling: Walk 45% + Bus 30% + Driven 20% = 95%. Cycling = 5%. 5% of 200 = 10 students cycle.
    4. Juice content: 12% of 375 mL = 45 mL of juice. For 60 mL: need 60 ÷ 0.12 = 500 mL minimum bottle size.