Percentages and Financial Applications — Solutions

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1. Calculate the new amount after each percentage change

   1. Increase $250 by 12%: $250 × 1.12 = $280
   2. Decrease $340 by 25%: $340 × 0.75 = $255
   3. Increase 480 kg by 7.5%: 480 × 1.075 = 516 kg
   4. Decrease $1200 by 30%: $1200 × 0.70 = $840
   5. Increase $85 by 110%: $85 × 2.10 = $178.50
   6. Decrease $620 by 15%: $620 × 0.85 = $527
   7. Increase 3500 by 4%: 3500 × 1.04 = 3640
   8. Decrease $90 by 33⅓%: Multiplier = 1 − 1/3 = 2/3. $90 × 2/3 = $60
   9. Increase $760 by 2.5%: $760 × 1.025 = $779
   10. Decrease 840 g by 12.5%: 840 × 0.875 = 735 g

2. Find the original value before the percentage change

   1. After 20% increase, price is $180: Original = $180 ÷ 1.20 = $150
   2. After 10% decrease, salary is $45 000: Original = $45 000 ÷ 0.90 = $50 000
   3. Phone costs $660 incl. GST: Pre-GST = $660 ÷ 1.10 = $600
   4. After 35% discount, coat costs $117: Original = $117 ÷ 0.65 = $180
   5. Population increased 8% to 32 400: Original = 32 400 ÷ 1.08 = 30 000
   6. Car sold for $18 700 at 15% loss: Original = $18 700 ÷ 0.85 = $22 000

3. Financial applications

   1. Plumber charges $350 + GST: $350 × 1.10 = $385
   2. Laptop $1320 incl. GST, find GST component: Pre-GST = $1320 ÷ 1.10 = $1200. GST = $1320 − $1200 = $120
   3. Vase bought $40, sold $58: Profit = $58 − $40 = $18. Profit% = (18/40) × 100 = 45%
   4. Car bought $25 000, sold $19 000: Loss = $6000. Loss% = (6000/25 000) × 100 = 24%
   5. Commission 3.5% on $28 000 sales: $28 000 × 0.035 = $980
   6. 2% commission, received $1450: Sales = $1450 ÷ 0.02 = $72 500
   7. Shirt wholesale $35, marked up 40%: Selling price = $35 × 1.40 = $49
   8. Textbook $55 incl. GST, student gets 10% off pre-GST then pays GST: Pre-GST = $55 ÷ 1.10 = $50. After 10% discount: $50 × 0.90 = $45. Add GST: $45 × 1.10 = $49.50

4. Multi-step percentage problems

   1. Price up 20% then down 20%: If P = $100: after 20% up → $120. After 20% down: $120 × 0.80 = $96. Not the same — the final price is 96% of the original. The 20% decrease is applied to a larger amount.
   2. Restaurant bill $120, +10% service then −5% loyalty: $120 × 1.10 = $132. $132 × 0.95 = $125.40
   3. $500 TV, 30% off then 10% off sale price: After 30%: $500 × 0.70 = $350. After 10%: $350 × 0.90 = $315. Overall saving: $500 − $315 = $185. % saving = (185/500) × 100 = 37%
   4. Tom earns $3200, +5% then +3%: After 5%: $3200 × 1.05 = $3360. After 3%: $3360 × 1.03 = $3460.80

5. Real-world financial problems

   1. House $480 000 to $552 000 over 3 years: Total % increase = (72 000/480 000) × 100 = 15%. For equal annual growth: (1 + r)³ = 1.15 → r = 1.15^(1/3) − 1 ≈ 0.0477 ≈ 4.77% per year
   2. $299 incl. GST, 15% off for members: The $299 already includes GST. A 15% discount on $299 = $299 × 0.85 = $254.15. Note: the discount applies to the GST-inclusive price here. If discount applied pre-GST: pre-GST = $299 ÷ 1.10 ≈ $271.82; 15% off → $231.05; then GST: $231.05 × 1.10 ≈ $254.15. Same result because multiplication is commutative.
   3. 200 units at $18 each, 180 sold at $28, 20 at 40% discount: Total cost = 200 × $18 = $3600. Revenue from 180 units: 180 × $28 = $5040. Discounted price: $28 × 0.60 = $16.80. Revenue from 20 units: 20 × $16.80 = $336. Total revenue = $5040 + $336 = $5376. Profit = $5376 − $3600 = $1776. Profit % = (1776/3600) × 100 = 49.3%

6. Salary negotiations

   1. $58 000 × 1.04 = $60 320
   2. $60 320 × 1.03 = $62 129.60
   3. 1.04 × 1.03 = 1.0712. Single equivalent increase = 7.12%

7. Real estate commission

   1. $420 000 × 0.025 = $10 500; $675 000 × 0.025 = $16 875; $310 000 × 0.025 = $7 750
   2. $10 500 + $16 875 + $7 750 = $35 125
   3. $35 125 − $4 200 = $30 925 profit

8. Comparing deals

   1. $500 × 0.80 × 0.90 = $360
   2. $500 × 0.72 = $360
   3. Both stores charge $360 — no difference.
   4. The second 10% applies to the reduced price ($400), not $500. 0.80 × 0.90 = 0.72 → exactly 28% off, matching Store B. Successive discounts compound rather than add.

9. Property investment

   1. ($552 000 − $480 000) ÷ $480 000 × 100 = 15%
   2. (1 + r)³ = 1.15; r = 1.15⅓ − 1 ≈ 4.77% per year
   3. $552 000 × (1.0477)&sup4; ≈ $552 000 × 1.2049 ≈ $665 100

10. Business profit analysis

    1. 200 × $18 = $3 600
    2. 180 × $28 = $5 040
    3. $28 × 0.60 = $16.80 per unit; 20 × $16.80 = $336
    4. Total revenue = $5 376. Profit = $5 376 − $3 600 = $1 776
    5. (1 776 ÷ 3 600) × 100 = 49.3%