Repayments, outstanding balances and total interest | Year 12

01 / LEARN

The idea, step by step

Use the same date for both parts of the balance

After $k$ payments, grow the original principal to time $k$ and subtract the accumulated effect of the $k$ payments. This is the closed-form version of the loan recurrence. It assumes the same timing and rate throughout.

Alternatively, if the exact repayment was chosen to exhaust the loan in $n$ periods, the balance after $k$ payments is the present value of the remaining $n$ − $k$ repayments. Both methods should agree.

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

The last payment is usually smaller

A chosen cent-valued repayment may not divide the loan into an exact number of periods. Find the first period in which a full payment would make the balance nonpositive. Pay only the opening balance plus the final period’s interest.

Total interest equals all actual payments minus the original principal, when there are no fees. Do not multiply the usual repayment by a rounded period count if the final payment is reduced. State whether regular repayments were rounded to cents and retain unrounded balances unless the question specifies a cent-by-cent schedule.

A common trap

Counting a full final repayment after the debt has already been nearly cleared.

Instead: Calculate the last amount as the previous balance times 1+ $i$ ; do not overpay the debt.

02 / WORKED EXAMPLES

See the thinking

Example 1

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.

Use $B_{12} = 15000 {(1.005)}^{12} - 500 ({(1.005)}^{12} - 1) / 0.005$ .
This agrees with 12 recurrence steps.

AUD 9,757.39

Example 2

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.

Apply $B_{next} = 1.01 B - 400$ until the next full payment would be too large.
The sixth amount is the remaining balance grown one more month.
Total interest is five regular payments of AUD 400 plus the final payment, minus the AUD 2,000 principal.

6 payments; final AUD 62.23; interest AUD 62.23.

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

AUD 216.99

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.

03 / WORKSHEET

Your turn

6 questions · 18 marks · Work at your own pace.

Worksheet PDF 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 $700 (1 - {1.004}^{- 9}) / 0.004$ .
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 ↗

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