IB Mathematics SL 1.7 Amortization and annuities using technology AI SL Paper 1- Exam Style Questions- New Syllabus
Question
Emma plans to invest money for 10 years in an account paying 3.5% interest, compounded annually. She anticipates the annual inflation rate to be 2% per year throughout the 10-year period. Emma would like her investment to be worth a real value of \$4000, compared to current values, at the end of the 10-year period. She is considering two options.
Option 1: Make a one-time investment at the start of the 10-year period.
Option 2: Invest \$1000 at the start of the 10-year period and then invest \$x into the account at the end of each year (including the first and last years).
(a) For Option 1, calculate the minimum amount Emma would need to invest. Give your answer to the nearest dollar. [3]
(b) For Option 2, obtain the minimum value of \( x \) that Emma would need to invest each year. Give your answer to the nearest dollar. [3]
▶️ Answer/Explanation
Markscheme
(a) METHOD 1
Settings: \( N = 10 \), \( I = 1.5 \), \( FV = 4000 \), \( P/Y = 1 \), \( C/Y = 1 \). A1 M1 \[ PV \approx 3447 \] Answer: \$3447 A1
METHOD 2
\[ 4000 \times 1.02^{10} = PV \times 1.035^{10} \] \[ PV \approx \frac{4000 \times 1.02^{10}}{1.035^{10}} \approx 3456.6667 \] Answer: \$3457 A1 M1 A1
METHOD 3
Real rate: \( \frac{1.035}{1.02} \approx 1.01470588235 \). \[ \begin{aligned} 4000 &= PV \times 1.01470588235^{10} \\ PV &\approx \frac{4000}{1.01470588235^{10}} \approx 3456.6667 \end{aligned} \] Answer: \$3457 A1 M1 A1
[3 marks]
(b) METHOD 1
Settings: \( N = 10 \), \( I = 3.5 \), \( PV = -3446.66 \) (Method 1) or \( -3456.67 \) (Methods 2, 3), \( P/Y = 1 \), \( C/Y = 1 \). M1 \[ FV \approx 4861.87 \text{ or } 4875.97 \] Payment: \$294 or \$295 A1 A1
METHOD 2
Settings: \( N = 10 \), \( I = 3.5 \), \( PV = -1000 \), \( FV = 4875.977 \), \( P/Y = 1 \), \( C/Y = 1 \). A1 M1 \[ \begin{aligned} 1000 \times 1.035^{10} &\approx 1410.5989 \\ x \times \frac{1.035^{10} – 1}{0.035} &\approx x \times 11.7313966789 \\ 1410.5989 + x \times 11.7313966789 &\approx 4875.9777 \\ x &\approx 295.384 \end{aligned} \] Answer: \$295 A1
Answer: \$295 [3 marks]
Total Marks: 6