Repayments, outstanding balances and total interest | Year 12

01 / LEARN

Understand the idea.

Use the same date for both parts of the balance

An outstanding balance is measured at a particular payment date. Just after payment $k$ , grow the original principal through $k$ periods and subtract the accumulated effect of those $k$ payments: $B_{k} = P {(1 + i)}^{k} - R \frac{{(1 + i)}^{k} - 1}{i}$ .

The last included payment has exponent zero because it has just occurred. If the exact original payment clears the loan after $n$ periods, the same balance is the present value of the remaining $n - k$ payments. These are two views of the same dated debt.

B_{k} = P {(1 + i)}^{k} - R \frac{{(1 + i)}^{k} - 1}{i}

The last payment is usually smaller

With a chosen payment rounded to cents, the final debt will usually be less than a full regular payment. Continue until the next full payment would overpay; the actual final payment is the previous balance plus that final period’s interest.

With no fees, total interest is the sum of actual payments minus the original principal. Include the reduced final payment rather than multiplying a rounded payment count by the usual amount. Retain unrounded running balances unless a different rounding convention is specified.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Build the model

AUD 15,000 is borrowed at 0.5% monthly interest, with AUD 500 month-end repayments. Find the balance after 12 payments.

Worked solution

1
Identify the matching monthly quantities.
$P = 15000, i = 0.005, R = 500, k = 12$
The required balance is immediately after the twelfth month-end payment.
2
Subtract accumulated repayments from accumulated principal.
$B_{12} = 15000 {(1.005)}^{12} - 500 \frac{{(1.005)}^{12} - 1}{0.005}$
Both components are valued at month twelve, and the twelfth payment has no extra growth.
3
Evaluate without intermediate rounding.
$B_{12} \approx 9757.385992 \approx AUD 9757.39$
The result is debt remaining after the payments, not the total amount already paid.

Answer

AUD 9,757.39

EXAMPLE 02

Use the model to make a decision

A loan is AUD 2,000 at 1% monthly interest, repaid by AUD 400 at each month end. Find the number of payments, final payment and total interest.

Worked solution

1
Apply the monthly recurrence through five regular payments.
$\begin{aligned} B_{1} & = 2000 (1.01) - 400 = 1620 \\ B_{2} & = 1620 (1.01) - 400 = 1236.20 \\ B_{3} & = 1236.20 (1.01) - 400 = 848.562 \end{aligned}$
Each interest charge uses the outstanding balance before that month’s repayment.
2
Find the balance after payments four and five.
$\begin{aligned} B_{4} & = 848.562 (1.01) - 400 = 457.04762 \\ B_{5} & = 457.04762 (1.01) - 400 = 61.6180962 \end{aligned}$
The loan still has a positive balance after the fifth AUD 400 payment.
3
Calculate the reduced sixth payment.
$F = 61.6180962 (1.01) = 62.234277162 \approx AUD 62.23$
Only the remaining debt and its final month’s interest are payable, so a sixth full AUD 400 payment is unnecessary.
4
Subtract principal from the actual total paid.
$\begin{aligned} I & = 5 (400) + 62.234277162 - 2000 \\ \approx AUD 62.23 \end{aligned}$
There are six payments altogether; five are regular and the last is reduced.

Answer

6 payments; final AUD 62.23; interest AUD 62.23.

03 / YOUR TURN

Try it yourself.

Check your understanding

The balance before a final month is AUD 215.70. The monthly rate is 0.6%. Find the final payment.

Show answer & reasoning

AUD 216.99

The final payment includes the last month’s interest on AUD 215.70: $215.70 (1.006) = 216.9942$ . Rounding that final amount to cents gives AUD 216.99.

Keep practising

6 worksheet questions · 18 marks

Open worksheet Worked solutions

Read the questions online

A debt of AUD 7,000 has been reduced to AUD 2,100 after total repayments of AUD 5,600. Find the interest charged so far.
3 marks
The balance before a final month is AUD 215.70. The monthly rate is 0.6%. Find the final payment.
3 marks
A no-fee loan of AUD 8,000 has 20 payments of AUD 430 and a final payment of AUD 195. Find the total interest.
3 marks
For AUD 5,000 at 0.75% monthly and AUD 250 repayments, calculate the debt after 6 payments.
3 marks
A payment schedule has 11 usual payments of AUD 300 followed by AUD 184.20. Its principal is AUD 3,200. Find total interest.
3 marks
An exact annuity repayment leaves 9 payments of AUD 700 at an interest rate of 0.4% per period. Find the outstanding balance immediately after the most recent payment.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

A debt of AUD 7,000 has been reduced to AUD 2,100 after total repayments of AUD 5,600. Find the interest charged so far.
3 marks
Principal repaid is $7000 - 2100 = 4900$ .
Interest paid is $5600 - 4900 = 700$ .
AUD 700.00
The balance before a final month is AUD 215.70. The monthly rate is 0.6%. Find the final payment.
3 marks
Add the final month’s interest to the remaining balance.
$215.70 \times 1.006 = 216.9942,$ so pay AUD 216.99 under final-cent rounding.
AUD 216.99
A no-fee loan of AUD 8,000 has 20 payments of AUD 430 and a final payment of AUD 195. Find the total interest.
3 marks
Total payments is $20 \times 430 + 195 = 8795$ .
Interest is $8795 - 8000 = 795$ .
AUD 795.00
For AUD 5,000 at 0.75% monthly and AUD 250 repayments, calculate the debt after 6 payments.
3 marks
Use the outstanding-balance formula with $k = 6$ .
Retain unrounded intermediate values.
AUD 3,700.85
A payment schedule has 11 usual payments of AUD 300 followed by AUD 184.20. Its principal is AUD 3,200. Find total interest.
3 marks
Total paid is $11 \times 300 + 184.20 = 3484.20$ .
Subtract principal to obtain AUD 284.20.
AUD 284.20
An exact annuity repayment leaves 9 payments of AUD 700 at an interest rate of 0.4% per period. Find the outstanding balance immediately after the most recent payment.
3 marks
Discount the remaining nine payments to the current date.
Balance is $\frac{700 (1 - {1.004}^{- 9})}{0.004}$ .
AUD 6,175.83

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 General Mathematics financial models: investments and annuities, superannuation, income streams and borrowing. Original practice uses supplied hypothetical rates, taxes and fees; statutory or commercial conditions are not inferred.

Official curriculum source ↗

In this lesson

Understand the idea.

Use the same date for both parts of the balance

The last payment is usually smaller

See the method in action.

Build the model

Identify the matching monthly quantities.

Subtract accumulated repayments from accumulated principal.

Evaluate without intermediate rounding.

Use the model to make a decision

Apply the monthly recurrence through five regular payments.

Find the balance after payments four and five.

Calculate the reduced sixth payment.

Subtract principal from the actual total paid.

Try it yourself.

Check your understanding

Keep practising