Chapter 21: Percentages, Percentage Change, Interest, Growth and Decay > HARD QUESTIONS (Questions 15-20)
HARD QUESTIONS (Questions 15-20)
Question 1 of 61 / 6
Compound Interest formula: A = P(1 + r)^n Where A = Final amount, P = Principal, r = Rate (decimal), n = Number of years P = $5000 r = 6% = 0.06 n = 3 years A = 5000 × (1 + 0.06)³ A = 5000 × (1.06)³ A = 5000 × 1.191016 A = $5955.08 (to 2 decimal places) Therefore, the investment is worth $5955.08 after 3 years.This is exponential growth with multiplier 1.25 per hour. Formula: Final = Initial × (multiplier)^time Initial = 800 Multiplier = 1 + 0.25 = 1.25 Time = 4 hours Final = 800 × (1.25)⁴ Step by step: (1.25)¹ = 1.25 (1.25)² = 1.5625 (1.25)³ = 1.953125 (1.25)⁴ = 2.44140625 Final = 800 × 2.44140625 = 1953.125 Rounding to whole bacteria: 1953 Therefore, there will be approximately 1953 bacteria after 4 hours.This is exponential decay with multiplier 0.85 per year. Formula: Final = Initial × (multiplier)^time Initial = 200g Multiplier = 1 - 0.15 = 0.85 Time = 5 years Final = 200 × (0.85)⁵ Calculate (0.85)⁵: (0.85)¹ = 0.85 (0.85)² = 0.7225 (0.85)³ = 0.614125 (0.85)⁴ = 0.52200625 (0.85)⁵ = 0.4437053125 Final = 200 × 0.4437053125 = 88.74g (to 2 dp) Therefore, 88.74g of the substance remains after 5 years.Part 1: Calculate the final price with successive discounts. Original price = $120 After 30% off: $120 × 0.70 = $84 After extra 10% off: $84 × 0.90 = $75.60 The customer pays $75.60. Part 2: Show this is different from 40% off. If 40% off was taken: $120 × 0.60 = $72 Comparison: Successive discounts (30% then 10%): $75.60 Single 40% discount: $72.00 Difference: $75.60 - $72.00 = $3.60 Explanation: The 10% is taken off the already reduced price ($84), not the original price ($120). So the second discount is smaller than expected. Combined multiplier: 0.70 × 0.90 = 0.63, which is a 37% total discount, not 40%.Part (a): Future value (4 years from now) Growth multiplier = 1.08 Future value = Current value × (1.08)^4 Future value = 25000 × (1.08)^4 (1.08)^4 = 1.36048896 Future value = 25000 × 1.36048896 = $34,012 (to nearest dollar) Part (b): Past value (2 years ago) If the value increased by 8% per year to reach $25,000 now: Current value = Past value × (1.08)^2 25000 = Past value × (1.08)^2 25000 = Past value × 1.1664 Past value = 25000 ÷ 1.1664 Past value = $21,433 (to nearest dollar) Answers: (a) $34,012 in 4 years, (b) $21,433 two years ago.Part (a): Bank A - Simple Interest at 5% Simple Interest: I = PRT I = 8000 × 0.05 × 5 = $2000 Final amount = Principal + Interest = $8000 + $2000 = $10,000 Part (b): Bank B - Compound Interest at 4.5% Compound Interest: A = P(1 + r)^n A = 8000 × (1.045)^5 Calculate (1.045)^5: (1.045)^1 = 1.045 (1.045)^2 = 1.092025 (1.045)^3 = 1.141166 (1.045)^4 = 1.192519 (1.045)^5 = 1.246182 A = 8000 × 1.246182 = $9969.46 Part (c): Comparison Bank A: $10,000.00 Bank B: $9,969.46 Bank A gives the better return by $10,000 - $9,969.46 = $30.54 Even though Bank A has a lower interest rate for compound interest calculations, the simple interest at 5% beats compound interest at 4.5% over 5 years in this case.
