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Number and Algebra - Loans and mortgages

Grade 11IB_AI

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Loans are typically repaid using the reducing balance method, where interest is calculated on the remaining principal at the end of each period.

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The TVM (Time Value of Money) solver on a GDC is the primary tool for these calculations. The variables are: NN (total number of payment periods), I%I\% (annual interest rate), PVPV (present value or initial loan amount), PMTPMT (periodic payment amount), FVFV (future value, which is 00 when the loan is fully repaid), P/YP/Y (payments per year), and C/YC/Y (compounding periods per year).

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For PVPV and PMTPMT signs: If you receive the loan, PVPV is positive (money into your pocket). Payments made to the bank (PMTPMT) are negative (money out of your pocket).

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Total interest paid is calculated as the difference between the total amount paid over the life of the loan and the original principal borrowed: Total Interest=(N×∣PMT∣)−PV\text{Total Interest} = (N \times |PMT|) - PV.

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The outstanding balance of a loan at any time tt is the FVFV calculated by setting NN to the number of payments already made.

📐Formulae

A=P(1+r100k)knA = P\left(1 + \frac{r}{100k}\right)^{kn}

Total Repaid=n×PMT\text{Total Repaid} = n \times PMT

Interest Paid=(n×PMT)−PV\text{Interest Paid} = (n \times PMT) - PV

Outstanding Balance=Future Value after n payments\text{Outstanding Balance} = \text{Future Value after } n \text{ payments}

💡Examples

Problem 1:

Sarah takes out a mortgage of EUR 200000 at an annual interest rate of 4%4\% compounded monthly. The loan is to be repaid over 25 years with equal monthly installments. Calculate the monthly payment.

Solution:

Using the TVM solver: N=25×12=300N = 25 \times 12 = 300 I%=4I\% = 4 PV=200000PV = 200000 FV=0FV = 0 P/Y=12P/Y = 12 C/Y=12C/Y = 12 Solving for PMTPMT, we get PMT=−1055.67PMT = -1055.67.

Explanation:

The monthly payment is EUR 1055.67. We use N=300N = 300 because there are 300 months in 25 years. PVPV is positive because Sarah receives the money.

Problem 2:

Using the details from the previous example (Loan: EUR 200000, PMTPMT: EUR 1055.67, NN: 300), calculate the total interest paid over the 25 years.

Solution:

Total amount paid: 300×1055.67=316701300 \times 1055.67 = 316701 Interest calculation: 316701−200000116701\begin{array}{r} 316701 \\ - 200000 \\ \hline 116701 \end{array} Total Interest = EUR 116701.

Explanation:

The total interest is the total of all payments minus the original principal borrowed. We use vertical subtraction for clarity.

Problem 3:

A car loan of USD 15000 is taken at 6%6\% interest per annum, compounded quarterly. If the quarterly payments are USD 1000, find the number of full quarters required to pay off the loan.

Solution:

Using the TVM solver: I%=6I\% = 6 PV=15000PV = 15000 PMT=−1000PMT = -1000 FV=0FV = 0 P/Y=4P/Y = 4 C/Y=4C/Y = 4 Solving for NN, we get N≈16.99N \approx 16.99.

Explanation:

Since NN represents the number of quarters, it will take 17 quarters to fully pay off the loan (the final payment will be slightly less than USD 1000).