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9.04 Loans

Worksheet
Loan repayments
1

Iain takes out a loan to purchase a jetski. He makes 15 equal loan repayments of \$5239. Calculate the total amount paid back on the loan.

2

Dylan takes out a loan to purchase a property. He makes equal monthly loan repayments of \$4600 over 27 years to pay it off. Calculate the total amount paid back on the loan.

3

Gwen takes out a loan to purchase a surround sound system. She makes 11 equal loan repayments. The total loan amount paid back is \$6600.

Calculate the amount of each repayment.

4

Lisa takes out a loan to purchase a small boat. She pays it back in equal monthly repayments over 6 years. The total loan amount paid back is \$55\,800.

Calculate the amount of each repayment.

5

Jimmy takes out a loan of \$700 to purchase a computer. The loan earns interest at 6\% p.a. compounded annually. Repayments of \$140 are made annually.

a

Calculate the amount Jimmy still owes after his first repayment.

b

Calculate the amount Jimmy still owes after his second repayment.

6

Kathleen takes out a loan of \$88\,000 to renovate her home. The loan earns interest at 8\% p.a. compounded monthly. Repayments of \$11\,264 are made annually.

a

Calculate the amount Kathleen will still owe after the first repayment.

b

Calculate the amount Kathleen will still owe after the second repayment.

7

Han received a 9-year \$43\,000 loan at 10\% p.a. monthly reducible interest. He makes monthly instalments of \$209.

a

Find the amount owing after:

i

1 month

ii

2 months

iii

3 months

b

Is the amount owing increasing or decreasing?

c

Will this loan eventually be paid off if the instalments remain the same? Explain your answer.

8

Joanne takes out a loan of \$54\,000 to purchase a plot of land. The loan earns interest at 2\% p.a. compounded annually. Repayments of \$864 are made annually.

a

How much interest is added to Joanne's loan in the first year?

b

Calculate the amount Joanne still owes after the first repayment.

c

Should Joanne continue to make annual payments of \$864? Explain your answer.

d

Suggest a better annual payment amount for Joanne's loan.

9

Luke takes out a loan of \$73\,200 to start a business. The loan earns interest at 8\% p.a. compounded annually. State whether the loan will be paid off in 3 years if the following amounts are repaid monthly:

a

\$1627

b

\$2034

c

\$2562

10

Valentina is deciding between two \$109\,000 home loans. She has the capacity to pay \$3400 per month.

  • Option 1: 3.2\% p.a. over 3 years with fixed monthly repayments of \$3179.

  • Option 2: 2.6\% p.a. over 3 years with minimum monthly repayments of \$3151 that enables paying more than the minimum monthly repayment

a

What is the total repayment she will have to make with Option 1?

b

What is the maximum total repayment she will have to make with Option 2?

c

Which loan will cost less?

Reducing balance loan tables
11

Ivan takes out a car loan for \$24\,000. He is charged 8.1\% per annum interest, compounded monthly. Ivan makes repayments of \$450 at the end of each month.

Complete the following table which tracks the loan over the first three months:

MonthOpening BalanceInterestRepaymentClosing Balance
124\,00016245023\,712
2
3
12

A car loan of \$6000 earns interest at 8\% p.a. compounded annually. Repayments of \$720 are made annually.

Complete the following table which tracks the loan over the first three years:

\text{Time period} \\ (n)\text{Value at beginning} \\ \text{of time period}\text{Interest at end} \\ \text{of time period}\text{Repayment} \\ \text{this period}\text{Amount at} \\ \text{end of time period}
1\$6000
2\$460.80\$720
3\$5500.80\$720
13

A loan of \$54\,000 earns interest at 5.4\% p.a. compounded monthly. Repayments of \$1500 are made monthly.

Complete the following table which tracks the loan amount over the first three months:

\text{Time period} \\ (n)\text{Value at beginning} \\ \text{of time period}\text{Interest at end} \\ \text{of time period}\text{Repayment} \\ \text{this period}\text{Amount at} \\ \text{end of time period}
1\$243.00\$1500\$52\,743
2\$237.34
3\$51\,480.34\$231.66\$1500
14

Dave took out a loan of \$6500 to start his business on January 1. Interest on the loan is charged at 10\% p.a. from the time the loan is taken out. He made yearly repayments of \$2500 on December 31 each year.

a

Complete the repayment table:

YearBalance owing at beginning of yearInterest charged in this periodBalance owing at end of yearBalance owing after repayment
1
2
3
b

Find the amount of interest charged between his first and second repayment.

15

Monthly repayments of \$3990 are made on a loan of \$158\,800 borrowed at a rate of 12\% p.a. compounded monthly.

Consider the following table of home loan repayments:

\text{Month }(n)\text{Principal }(P)\text{Interest }(I)P + IP + I - R
1\$158\,800\$1588.00\$160\,388.00\$156\,398.00
2\$156\,398.00\$1563.98\$157\,961.98\$153\,971.98
3
a

What is the principal at the beginning of the third month?

b

Calculate the interest charged for the third month.

c

How much money is owed at the beginning of the fourth month?

16

The following table shows the principal and interest over the first 4 months of a loan:

\text{Month}PIP +IP + I - R
1\$20\,000\$50.00\$20\,050.00\$19\,150.00
2\$19\,150.00\$47.88\$19\,197.88\$18\,297.88
3\$18\,297.88\$45.74\$18\,343.62\$17\,443.62
4\$17\,443.62\$43.61\$17\,487.23\$16\,587.23
a

Find the value of each repayment, R.

b

What is the annual interest rate charged on the loan to the nearest percent?

c

Calculate the value of the principal at the start of month 6.

17

Kate takes out a personal loan for \$40\,000. The interest on the loan is charged quarterly and Kate makes repayments of \$900 at the end of each quarter.

Calculations for the first quarter are shown in the following table:

QuarterOpening BalanceInterestRepaymentClosing Balance
140\,00040090039\,500
2
3
a

Calculate the quarterly interest rate Kate is charged for this loan as a percentage.

b

State the annual interest rate of this loan as a percentage.

c

Calculate the values for Quarters 2 and 3 in the table.

d

How much of the loan has Kate repaid in the first three quarters of the year?

18

Maximilian takes out a mortgage to purchase an investment property. A number of his payments and balances are shown in the following table:

MonthOpening BalanceInterestRepaymentClosing Balance
1x3500417\,970
2417\,9701462.903500
3y
a

Calculate the monthly interest rate charged on this loan as a percentage. Round your answer to two decimal places.

b

Calculate the annual interest rate charged on this loan as a percentage. Round your answer to two decimal places.

c

Calculate the value of x, the initial amount borrowed for this mortgage.

d

Calculate the value of y in the table.

19

A study abroad loan of \$8500 earns interest which is compounded monthly. Repayments of \$2550 are made half yearly. The following table documents the repayment of the loan:

\text{Time period} \\ (n)\text{Value at beginning} \\ \text{of time period}\text{Interest at end} \\ \text{of time period}\text{Repayment} \\ \text{this period}\text{Amount at} \\ \text{end of time period}
1\$8500\$416.25\$2550\$6366.25
2\$6366.25\$311.76\$2550\$4128
3\$4128\$202.15\$2550\$1780.15
4\$1780.15\$87.17\$1867.33\$0

Calculate the total loan amount paid.

20

A small loan of \$4500 to pay for a holiday earns interest at 4\% p.a. compounded annually. Repayments of \$2000 are made annually.

a

Complete the following table which tracks the loan amount over three years:

\text{Time} \\ \text{period } (n)\text{Value at beginning} \\ \text{of time period}\text{Interest at end} \\ \text{of time period}\text{Repayment} \\ \text{this period}\text{Amount at} \\ \text{end of time period}
1\$4500\$2000
2\$107.20
3\$787.20\$0
b

Why is the repayment in the third year smaller than the other repayments?

c

Calculate the total loan amount paid.

21

You take out a personal loan of \$10\,000 at 11\% reducible p.a. The term of the loan is 3 years, and yearly repayments of \$2600 are made. The balance owing is paid at the end of 3 years.

a

Complete the loan repayment table:

Time periodBalance owing at beginning of periodBalance owing plus interest during periodRepaymentBalance owing at end of period
110\,0002600
22600
3
b

Find the total amount of interest charged on the loan.

c

Find the total repayment over 3 years.

22

A study abroad loan of \$13\,600 earns interest at 2.4\% p.a. compounded monthly. Repayments of \$4080 are made either half-yearly or yearly.

a

Complete the following table which tracks the repayment of the loan with half-yearly payments:

\text{Time} \\ \text{period }(n)\text{Value at beginning} \\ \text{of time period}\text{Interest at end} \\ \text{of time period}\text{Repayment} \\ \text{this period}\text{Amount at} \\ \text{end of time period}
1\$13\,600\$164.02\$4080
2\$9684.02\$116.79
3\$5720.81\$68.99\$4080
4\$0
b

What is the total loan amount paid if making half yearly repayments?

c

Complete the following table that tracks the repayment of the loan with yearly payments:

\text{Time} \\ \text{period }(n)\text{Value at beginning} \\ \text{of time period}\text{Interest at end} \\ \text{of time period}\text{Repayment} \\ \text{this period}\text{Amount at} \\ \text{end of time period}
1\$13\,600\$330.01\$8160
2\$5770.01\$140.01
d

What is the total loan amount paid if making yearly repayments?

e

What is the better payment option in order to reduce the total amount paid?

23

\$3700 is invested for three years at a rate of 7\% p.a., compounded annually.

a

Complete the table below to determine the final value of the investment:

Balance at beginning of yearInterest earned
First year\$3700\$259
Second year\$3959\$277.13
Third year
Fourth year-
b

Calculate the total interest earned over the three years.

24

Mr and Mrs Lyne have a \$520\,000 mortgage for their home. They are charged 5.16\% interest per annum, compounded monthly and make monthly repayments of \$3750.

a

Complete the table below, using the rounded answer to calculate the amounts for the following month:

\text{Month }\text{Opening Balance } (\$)\text{Interest }(\$)\text{Repayment }(\$)\text{Closing Balance }(\$)
1520\,00022363750518\,486
2
3
4
5
b

If they made fortnightly repayments of \$1\,875 instead of monthly repayments, what would be the result on the balance of their loan after 10 years?

25

A \$2000 loan is to be repaid at a reducible rate of 8\% p.a. There are two possible methods of repayment: repaying \$750 every 6 months or repaying \$375 every 3 months, and the interest is compounded at the same interval as payment.

a

Complete the repayment table, showing repayments made every 6 months:

Time periodBalance owing at beginning of periodBalance owing plus interest during periodRepaymentBalance owing at end of period
120002080750
2750
b

Complete the repayment table, showing repayments made every 3 months:

Time periodBalance owing at beginning of periodBalance owing plus interest during periodRepaymentBalance owing at end of period
1200020403751665
216651698.30375
3375
4375
c

Calculate the difference in the balances owing after 1 year.

26

A credit card charges annual interest of 18.24\%.

a

Calculate the monthly interest rate on this credit card, correct to two decimal places.

b

Roald owes \$3200 on his credit card and will pay it all off before using this card again. The table displays his payments:

MonthInterestRepaymentAmount Owing
3200
148.6775X
248.27753146.93
3Y753119.79
\ldots\ldots\ldots\ldots
\ldots\ldots\ldots\ldots
664.7675242.66
673.6975171.35
682.617598.96
691.517525.47
70ZW0
i

Calculate the value of X.

ii

Calculate the value of Y

iii

Calculate the value of Z.

iv

Calculate the value of W.

v

Calculate the total interest paid.

27

Xanthe takes out a car loan. The last few months of her repayments are shown in the following table:

MonthOpening BalanceInterestRepaymentClosing Balance
69868.484.34250622.82
70622.823.11250375.93
71375.931.88250127.81
72
a

Calculate the monthly interest rate charged on this loan, rounding your answer to one decimal place.

b

Calculate the values for the final row of the table, rounding your answers correct to two decimal places.

c

How many years did it take for Xanthe to pay off the loan?

d

Calculate her total repayments.

e

If her original loan was for \$15\,000, calculate the total interest paid on the loan.

28

Mr. and Mrs. Dave have a mortgage. The final months of their repayments are shown in the following table:

MonthOpening BalanceInterestRepaymentClosing Balance
14626\,452.84198.40500021\,651.24
14721\,651.24162.38500016\,813.62
14816\,813.62126.10500011\,939.72
14911\,939.7289.5550007029.27
1507029.2752.7250002081.99
151
a

Calculate the monthly interest rate charged on this loan. Round your answer to two decimal places.

b

Calculate the values for the final row of the table, rounding your answers to two decimal places.

c

How many years did it take for them to pay off the loan? Round your answer to two decimal places.

d

Calculate the total repayments.

e

If they paid \$302\,097.60 in interest, how much did they initially borrow?

Financial tables
29

The following financial table displays the monthly repayments on a \$1000 loan:

\text{Annual interest rate}5 \\ \text{years}10 \\ \text{years}15 \\ \text{years}20 \\ \text{years}25 \\ \text{years}
10\%21.2513.2210.759.659.09
11\%21.7413.7811.3710.329.80
12\%22.2414.3512.0011.0110.53
13\%22.7514.9312.6511.7211.28
14\%23.2715.5313.3212.4412.04
15\%23.7916.1314.0013.1712.81

Han received a 5-year \$151\,000 loan at 10\% p.a. monthly reducible interest. If the interest rate was increased to 15\% p.a., find the increase in the amount of each monthly instalment needed to be paid.

30

The following financial table displays the monthly repayments on a \$1000 loan:

\text{Annual interest rate}5 \\ \text{years}10 \\ \text{years}15 \\ \text{years}20 \\ \text{years}25 \\ \text{years}
6\%19.3311.108.447.166.44
7\%19.8011.618.997.757.07
8\%20.2812.139.568.367.72
9\%20.7612.6710.149.008.39
10\%21.2513.2210.759.659.09
11\%21.7413.7811.3710.329.80

Maria received a 5-year \$209\,000 loan at 10\% p.a. monthly reducible interest. If the interest rate fell to 6\% p.a., find the decrease in the amount of each monthly instalment needed to be paid.

31

The following financial table displays the monthly repayments on a \$1000 loan:

\text{Annual interest rate}10 \\ \text{years}15 \\ \text{years}20 \\ \text{years}25 \\ \text{years}30 \\ \text{years}
4\%10.127.406.065.284.77
5\%10.617.916.605.855.37
6\%11.108.447.166.446.00
7\%11.618.997.757.076.65
8\%12.139.568.367.727.34
9\%12.6710.149.008.398.05

Neil received a 10-year \$130\,000 loan at 6\% p.a. monthly reducible interest. If the interest rate was increased to 7\% p.a., find the increase in the total repayments needed to clear the debt.

32

ANZ offers the choice of two types of loans:

  • Option 1: 5.2\% p.a. flat rate interest over 25 years.

  • Option 2: 8.2\% p.a. reducible interest over 15 years.

The following financial table displays the monthly repayments on a \$1000 loan:

\text{Annual interest rate}15 \\ \text{years}20 \\ \text{years}25 \\ \text{years}30 \\ \text{years}35 \\ \text{years}
4.2\%9.067.676.836.285.88
5.2\%9.898.57.677.116.71
6.2\%10.729.338.57.947.55
7.2\%11.5610.179.338.788.38
8.2\%12.391110.179.619.21
9.2\%13.2211.831110.4410.05
a

Calculate the total repayments to be made with Option 1 on a \$170\,000 loan.

b

Calculate the total repayments to be made with Option 2 on a \$170\,000 loan.

c

Which loan will cost less in total?

33

Eileen is deciding between two 20-year loans to finance the purchase of a new \$5000 salon.

  • Option 1: 6\% p.a. reducible interest along with a one-off \$535 loan application fee.

  • Option 2: 8\% p.a. reducible interest along with a monthly account fee of \$41.

The following financial table displays the monthly repayments on a \$1000 loan:

\text{Annual interest rate}15 \\ \text{years}20 \\ \text{years}25 \\ \text{years}30 \\ \text{years}35 \\ \text{years}
4\%7.406.065.284.774.43
5\%7.916.605.855.375.05
6\%8.447.166.446.005.70
7\%8.997.757.076.656.39
8\%9.568.367.727.347.10
9\%10.149.008.398.057.84

Which loan should she choose? Explain your answer.

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Outcomes

ACMEM174

use technology and a recurrence relation to model a reducing balance loan

ACMEM175

investigate the effect of the interest rate and repayment amount on the time taken to repay a loan

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