(a) (i) Zak invests $500 at a rate of 2% per year simple interest.
Calculate the value of Zak’s investment at the end of 5 years.
(ii) Yasmin invests $500 at a rate of 1.8% per year compound interest.
Calculate the value of Yasmin’s investment at the end of 5 years.
(iii) Zak and Yasmin continue with these investments.
How many more complete years is it before the value of Yasmin’s investment is greater than the value of Zak’s investment?
(b) Xavier buys a car for $2500.
The value of the car decreases exponentially at a rate of 10% each year.
Calculate the value of Xavier’s car at the end of 5 years.
Give your answer correct to the nearest dollar.
(c) The number of a certain type of bacteria increases exponentially at a rate of r% each day.
After 22 days, the number of this bacteria has doubled.
Find the value of r.
▶️ Answer/Explanation
(a)(i) $550
Simple interest formula: Principal + (Principal × rate × time). Calculation: $500 + ($500 × 0.02 × 5) = $550.
(a)(ii) $546.65
Compound interest formula: Principal × (1 + rate)^time. Calculation: $500 × (1 + 0.018)^5 ≈ $546.65.
(a)(iii) 8 years
Zak’s investment grows linearly: $550 + ($10/year). Yasmin’s grows exponentially. After 13 total years (8 more), Yasmin’s $546.65 × (1.018)^8 ≈ $629.51 exceeds Zak’s $550 + $80 = $630.
(b) $1476
Exponential decay formula: Initial × (1 – rate)^time. Calculation: $2500 × (0.90)^5 ≈ $1476.22, rounded to nearest dollar.
(c) 3.2
Doubling means (1 + r/100)^22 = 2. Solving: r ≈ 100 × (2^(1/22) – 1) ≈ 3.20%.
(a) The value of Priya’s car decreases by 10% every year.
The value today is $7695.
(i) Calculate the value of the car after one year.
(ii) Calculate the value of the car one year ago.
(b) Ali invests $600 at a rate of 2% per year simple interest.
Calculate the value of Ali’s investment at the end of 5 years.
(c) Sara invests $500 at a rate of r% per year compound interest.
At the end of 12 years, the value of Sara’s investment is $601.35, correct to the nearest cent.
Find the value of r.
(d) The mass of a radioactive substance decreases exponentially at a rate of 3% each day.
(i) Find the overall percentage decrease at the end of 10 days.
(ii) Find the number of whole days it takes until the mass of this substance is one half of its original amount.
▶️ Answer/Explanation
(a)(i) $6925.50
After 10% decrease: $7695 × 0.90 = $6925.50.
(a)(ii) $8550
Let original value be x. Then x × 0.90 = 7695, so x = 7695 ÷ 0.90 = 8550.
(b) $660
Simple interest: 600 × 0.02 × 5 = $60. Total value: 600 + 60 = $660.
(c) r = 1.55
Using compound interest formula: 500(1 + r/100)¹² = 601.35. Solving gives r ≈ 1.55%.
(d)(i) 26.3%
Overall decrease: 100% – (0.97)¹⁰ × 100% ≈ 26.3%.
(d)(ii) 23 days
Solve (0.97)ⁿ = 0.5. Testing n=22 gives 0.506, n=23 gives 0.491, so 23 days needed.