Reducing-balance loans
Model reducing-balance loans using a recurrence relation; track repayments and interest using a table; compare loan options; analyse credit cards and interest-free periods.
Worked examples
Building a reducing-balance loan table
Straightforward
Problem
A loan has an opening balance of $6000. The monthly interest rate is 0.5% and the monthly repayment is $600. Complete the first 3 months of the loan table and state the closing balance at the end of month 3.
1
Set up the recurrence relation.
Each month:
Opening balance:
Opening balance:
2
Calculate month 1.
Interest
3
Calculate month 2.
Interest
4
Calculate month 3.
Interest
Answer
The closing balance at the end of month 3 is $4281.44.
Credit card with minimum repayments
Moderate
Problem
A credit card has an outstanding balance of $3000. Interest is charged at 21% per annum, compounded monthly. A minimum repayment of $60 per month is made. Find the balance at the end of 3 months and comment on what the result shows.
1
Find the monthly interest rate and set up the recurrence.
Monthly rate
2
Calculate month 1.
3
Calculate month 2.
4
Calculate month 3.
5
Comment on the result.
After 3 months of $60 repayments ($180 total), the balance has only reduced by:
Almost all of each repayment goes to interest, not the principal. The debt will take a very long time to repay at the minimum repayment level.
Almost all of each repayment goes to interest, not the principal. The debt will take a very long time to repay at the minimum repayment level.
Answer
Balance after 3 months: $2976.50. Minimum repayments barely dent the principal when the interest rate is high.
Comparing two loan options
Challenging
Problem
Two lenders offer loans of $12\,000. Lender A charges 6% p.a. (0.5% per month) with repayments of $700 per month. Lender B charges 7.2% p.a. (0.6% per month) with repayments of $680 per month. Find the closing balance after 2 months for each loan and identify which has the lower balance.
1
Set up the recurrence for each loan.
Lender A:
Lender B:
Lender B:
2
Calculate Lender A after 2 months.
Month 1:
Month 2:
Month 2:
3
Calculate Lender B after 2 months.
Month 1:
Month 2:
Month 2:
4
Compare the balances.
Lender A: $10\,716.80
Lender B: $10\,780.35
Lender A has the lower balance after 2 months, even though Lender B's repayment is smaller.
Lender B: $10\,780.35
Lender A has the lower balance after 2 months, even though Lender B's repayment is smaller.
Answer
After 2 months, Lender A balance: $10\,716.80; Lender B balance: $10\,780.35. Lender A (lower rate, higher repayment) reduces the loan faster.
Practise
Q1·Straightforward
A loan has an opening balance of $20\,000. The monthly interest rate is 0.5% and the monthly repayment is $500. Find the closing balance at the end of month 1.
Explanation
Q2·Straightforward
A loan has an opening balance of $8000. The monthly interest rate is 0.6%. Calculate the interest charged in the first month.
Explanation
Q3·Straightforward
A credit card has an outstanding balance of $1500. Interest is charged at 18% per annum, compounded monthly. No repayment is made. What is the balance at the end of one month?
Explanation
Monthly rate
Q4·Moderate
A loan has an opening balance of $10\,000. The monthly interest rate is 0.5% and the monthly repayment is $450. Find the closing balance at the end of month 2.
Explanation
**Month 1:**
**Month 2:**
**Month 2:**
Q5·Moderate
A loan is modelled by the recurrence relation with . Find the closing balance at the end of month 3, to the nearest cent.
Explanation
**Month 1:**
**Month 2:**
**Month 3:**
**Month 2:**
**Month 3:**
Q6·Moderate
A loan has an opening balance of $20\,000. The monthly interest rate is 0.5% and the monthly repayment is $800. Find the closing balance at the end of month 3, to the nearest cent.
Explanation
**Month 1:**
**Month 2:**
**Month 3:**
**Month 2:**
**Month 3:**
Q7·Moderate
A credit card purchase of $800 has an interest-free period that has now ended. Interest is then charged at 22% per annum, compounded monthly, with no repayment made. Find the outstanding balance after 2 months of interest charges, to the nearest cent.
Explanation
Monthly rate
Q8·Moderate
Two loans each have an opening balance of $15\,000. Loan A charges 6% per annum (compounded monthly); Loan B charges 5.4% per annum (compounded monthly). Both charge monthly repayments of $400. How much more interest does Loan A charge in the first month compared to Loan B?
Explanation
**Loan A** monthly rate:
**Loan B** monthly rate:
**Difference:**
**Loan B** monthly rate:
**Difference:**
Q9·Challenging
A home loan has an opening balance of $25\,000. The monthly interest rate is 0.75% and the monthly repayment is $1000. Find the closing balance at the end of month 3, to the nearest cent.
Explanation
**Month 1:**
**Month 2:**
**Month 3:**
**Month 2:**
**Month 3:**
Q10·Challenging
A loan has an opening balance of $3000. The monthly interest rate is 1% and the monthly repayment is $300. After how many months is the loan fully repaid? (The final payment may be less than $300.)
Explanation
Applying :
| Month | Balance |
|---|---|
| 0 | $3000.00 |
| 1 | $2730.00 |
| 2 | $2457.30 |
| 3 | $2181.87 |
| 4 | $1903.69 |
| 5 | $1622.73 |
| 6 | $1338.96 |
| 7 | $1052.35 |
| 8 | $762.87 |
| 9 | $470.50 |
| 10 | $175.21 |
| 11 | → loan cleared |
The loan is fully repaid in **11 months** (with a smaller final payment of $176.96).
Q11·Challenging
A personal loan has an opening balance of $5000. The monthly interest rate is 1% and the monthly repayment is $500. Calculate the total interest paid over the life of the loan, to the nearest cent.
Explanation
Tracking the loan (monthly rate 1%, repayment $500):
| Month | Opening | Interest | Closing |
|---|---|---|---|
| 1 | 5000.00 | 50.00 | 4550.00 |
| 2 | 4550.00 | 45.50 | 4095.50 |
| 3 | 4095.50 | 40.96 | 3636.46 |
| 4 | 3636.46 | 36.36 | 3172.82 |
| 5 | 3172.82 | 31.73 | 2704.55 |
| 6 | 2704.55 | 27.05 | 2231.59 |
| 7 | 2231.59 | 22.32 | 1753.91 |
| 8 | 1753.91 | 17.54 | 1271.45 |
| 9 | 1271.45 | 12.71 | 784.16 |
| 10 | 784.16 | 7.84 | 292.00 |
| 11 | 292.00 | 2.92 | Final payment: $294.92 |
Total paid
Total interest
*(Small rounding differences may appear when rounding to 2 d.p. at each step — use full precision in practice.)*
Total interest
*(Small rounding differences may appear when rounding to 2 d.p. at each step — use full precision in practice.)*
Q12·Challenging
A credit card has an outstanding balance of $2000. Interest is charged at 18% per annum, compounded monthly. Only the minimum repayment of $50 per month is made. Find the outstanding balance at the end of month 3, to the nearest cent.
Explanation
Monthly rate
**Month 1:**
**Month 2:**
**Month 3:**
Note: despite $150 in repayments, the balance has only reduced by $60.90 — most went to interest.
**Month 1:**
**Month 2:**
**Month 3:**
Note: despite $150 in repayments, the balance has only reduced by $60.90 — most went to interest.
Open Math
Reducing-balance loans
Financial Mathematics · MS-F4
Name:
Date:
Q1Straightforward
A loan has an opening balance of $20\,000. The monthly interest rate is 0.5% and the monthly repayment is $500. Find the closing balance at the end of month 1.
Q2Straightforward
A loan has an opening balance of $8000. The monthly interest rate is 0.6%. Calculate the interest charged in the first month.
Q3Straightforward
A credit card has an outstanding balance of $1500. Interest is charged at 18% per annum, compounded monthly. No repayment is made. What is the balance at the end of one month?
Q4Moderate
A loan has an opening balance of $10\,000. The monthly interest rate is 0.5% and the monthly repayment is $450. Find the closing balance at the end of month 2.
Q5Moderate
A loan is modelled by the recurrence relation with . Find the closing balance at the end of month 3, to the nearest cent.
Q6Moderate
A loan has an opening balance of $20\,000. The monthly interest rate is 0.5% and the monthly repayment is $800. Find the closing balance at the end of month 3, to the nearest cent.
Q7Moderate
A credit card purchase of $800 has an interest-free period that has now ended. Interest is then charged at 22% per annum, compounded monthly, with no repayment made. Find the outstanding balance after 2 months of interest charges, to the nearest cent.
Q8Moderate
Two loans each have an opening balance of $15\,000. Loan A charges 6% per annum (compounded monthly); Loan B charges 5.4% per annum (compounded monthly). Both charge monthly repayments of $400. How much more interest does Loan A charge in the first month compared to Loan B?
Q9Challenging
A home loan has an opening balance of $25\,000. The monthly interest rate is 0.75% and the monthly repayment is $1000. Find the closing balance at the end of month 3, to the nearest cent.
Q10Challenging
A loan has an opening balance of $3000. The monthly interest rate is 1% and the monthly repayment is $300. After how many months is the loan fully repaid? (The final payment may be less than $300.)
Q11Challenging
A personal loan has an opening balance of $5000. The monthly interest rate is 1% and the monthly repayment is $500. Calculate the total interest paid over the life of the loan, to the nearest cent.
Q12Challenging
A credit card has an outstanding balance of $2000. Interest is charged at 18% per annum, compounded monthly. Only the minimum repayment of $50 per month is made. Find the outstanding balance at the end of month 3, to the nearest cent.
Worked solutions and answers at openmath.au/year-12/standard-2/investments-and-loans/reducing-balance-loans