IB Mathematics SL 1.4 Financial applications of geometric sequences and series AI SL Paper 2 - Exam Style Questions - New Syllabus
Question
Tressy deposits \( 10\,000 \) CAD into a savings account that offers an annual interest rate of \( 3.4\% \), compounded monthly. She intends to use the total value of this investment after \( 6 \) years as a down payment for a house priced at \( 225\,000 \) CAD. The remaining balance will be financed through a bank loan.
(a) Calculate the total value of Tressy’s investment at the end of the \( 6 \)-year period.
(b) Determine the principal amount Tressy needs to borrow for her house loan.
The loan has a term of \( 15 \) years with an interest rate of \( 6.4\% \) per annum, compounded half-yearly. Tressy makes equal monthly repayments.
(c) Calculate the required monthly repayment for this loan.
After making payments for exactly \( 2 \) years, Tressy decides to increase her monthly repayment to \( 2\,200 \) CAD.
(d) (i) Calculate the total reduction in the loan principal over the following \( 3 \) full years of these increased payments.
(ii) Determine the total amount of interest Tressy will pay during this specific \( 3 \)-year period.
Most-appropriate topic codes (IB Mathematics AI SL 2025):
• SL 1.7: Amortization of loans and annuities using technology — parts (c), (d)
• SL 1.7: Calculation of interest paid and remaining balance — part (d)
▶️ Answer/Explanation
(a)
Initial investment \(PV = 10000\) CAD, rate \(3.4\%\) p.a. compounded monthly, \(n = 6 \times 12 = 72\).
Using compound interest formula:
\(FV = 10000 \left(1 + \frac{0.034}{12}\right)^{72}\)
\(\boxed{12259 \text{ CAD}}\) (to nearest dollar)
(b)
House price = 225000 CAD, initial payment = 12259 CAD.
Loan = \(225000 – 12259 = 212741\) CAD.
\(\boxed{212741 \text{ CAD}}\)
(c)
Loan amount \(PV = 212741\), term = 15 years, rate \(6.4\%\) p.a. compounded half-yearly, monthly payments.
Using financial solver:
\(N = 180\), \(I\% = 6.4\), \(PV = 212741\), \(FV = 0\), \(P/Y = 12\), \(C/Y = 2\).
Solve for PMT.
\(\boxed{1832 \text{ CAD}}\) (to nearest dollar)
(d)(i)
After 2 years, remaining balance = 194572 CAD.
New PMT = 2200, \(N = 36\).
Find FV after 36 payments.
Decrease = \(194572 – FV\).
\(\boxed{46473 \text{ CAD}}\) (to nearest dollar)
(d)(ii)
Total paid over 3 years = \(2200 \times 36 = 79200\) CAD.
Principal reduction = 46473 CAD.
Interest paid = \(79200 – 46473 = 32727\) CAD.
\(\boxed{32727 \text{ CAD}}\)