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IB Mathematics AI HL - Popular Quizzes

Amortization (Loans)

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Question 1

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[Maximum mark: 6]

In this question give all answers correct to two decimal places.

Stella receives a loan of € 3200032\hspace{0.15em}000 for her flower shop business at an interest rate 5.295.29 % per year, compounded monthly. She agrees to pay back the loan in 6060 equal installments, made at the end of each month over the next five years.

  1. Calculate the amount of monthly installment. [3]

Four years after she starts repaying the loan, Stella decides to repay the rest in a final single installment.

  1. Calculate the amount owing at the end of the four years. [3]

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Question 2

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[Maximum mark: 6]

Olivia takes a mortgage (loan) of $250000\$250\hspace{0.15em}000 to buy an apartment in Sydney. Interest\text{Interest} on the loan accumulates at the rate of $3.49\$3.49 % per year, compounded semi-annually. Olivia agrees with the bank to amortise the loan in monthly payments, made at the beginning of each month.

  1. Given that the loan is to be amortised over 3030 years, find:

    1. the monthly payment amount;

    2. the total amount paid in amortising the loan. [4]

  2. Olivia has the capacity to increase her monthly payments by $85\$85. Justify to Olivia why this may be a smart financial choice. [2]

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Question 3

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[Maximum mark: 5]

Tom takes out a loan of $80000\$80\hspace{0.15em}000 to purchase some new machinery for his farming business. He agrees to pay the bank $1200\$1200 at the end of every month to amortise the loan. Interest accumulates on the balance at a rate of 5.655.65 % per year, compounded monthly.

  1. Find the number of years and months it takes to pay back the loan. [2]

  2. Calculate the total amount that Tom pays in amortising the loan. [1]

  3. Tom decides to increase the monthly payment to $1500\$1500. How much interest will Tom save in comparison to the former payment schedule.[2]

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Question 4

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[Maximum mark: 19]

Nathan receives a lump sum inheritance of $55 000\$ 55\,000 and invests the money into a savings account with an annual interest rate of 7.5%7.5 \%, compounded quarterly.

  1. Calculate the value of Nathan's investment after 5 years, rounding your answer to the nearest dollar. [3]

After mm months, the amount in the savings account has increased to more than $70 000\$70\,000.

  1. Find the minimum value of mm, where m∈Nm\in N.[4]

Nathan is saving to purchase a property. The price of the property is $150 000\$150\,000. Nathan puts down a 15%15\% deposit and takes out a loan for the remaining amount.

  1. Write down the loan amount.[1]

The loan duration is for eight years, compounded monthly, with equal monthly payments of $1500\$1500 made by Nathan at the end of each month.

  1. For this loan, find
    1. the amount of interest paid by Nathan over the life of the loan.

    2. the annual interest rate of the loan, correct to two decimal places. [5]

After 55 years of paying this loan, Nathan decides to pay the outstanding loan amount in one final payment.

  1. Find the amount of the final payment after 55 years, rounding your answer to the nearest dollar. [3]

  2. Find the amount Nathan saved by making this final payment after 55 years, rounding your answer to the nearest dollar.[3]

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