Personal loans and comparing borrowing with saving | Year 11

01 / LEARN

Understand the idea.

Track a reducing balance

A reducing-balance loan calculates each period's interest on its opening balance. For an end-of-period repayment, first grow that balance by $1 + i$ , then subtract the repayment: $B_{next} = B (1 + i) - R$ . Reversing this order changes the interest charged.

A repayment reduces the balance only if it exceeds that period's interest. Keep full working precision unless the supplied model rounds every interest charge. A balance table helps show the opening amount, interest, repayment and closing amount for each period.

Monthly interest is added before the repayment
Month	Opening (AUD)	Interest (AUD)	Repayment (AUD)	Closing (AUD)
1	1000.00	10.00	200.00	810.00
2	810.00	8.10	200.00	618.10

Compare two decisions

Comparing borrowing now with saving first requires complete outlays and dates. Borrowing may provide earlier access but add interest and fees; waiting may change the future purchase price. Use the supplied later price rather than automatically assuming today's cash price still applies.

Explain what the numerical difference measures. A comparison of purchase outlays alone does not price earlier use, uncertain earnings or other omitted cash flows. State the cheaper option under the stated model without treating that limited calculation as a complete personal borrowing decision.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example

A AUD 1000 loan charges 1% monthly. A AUD 200 repayment is made at each month end. Find the balances after months 1 and 2.

Worked solution

1
Calculate the first month's interest.
$I_{1} = 1000 (0.01) = AUD 10$
The first month's opening balance is AUD 1000.
2
Subtract the first end-of-month payment.
$B_{1} = 1000 + 10 - 200 = AUD 810.00$
The payment occurs after the month's interest is added.
3
Calculate the second month's interest on the new balance.
$I_{2} = 810 (0.01) = AUD 8.10$
The second month starts with the reduced AUD 810 balance.
4
Subtract the second payment.
$B_{2} = 810 + 8.10 - 200 = AUD 618.10$
Applying the same end-of-month timing gives the second closing balance.

Answer

AUD 810.00; AUD 618.10

EXAMPLE 02

Worked example

An item costs AUD 1500 now. Borrowing costs 18 monthly payments of AUD 90 and a AUD 30 fee. Saving first leads to a cash price of AUD 1560, paid entirely from savings. Compare purchase outlays.

Worked solution

1
Find the full borrowing outlay.
$C_{borrow} = 18 (90) + 30 = AUD 1650$
The borrowing option requires all eighteen monthly payments and the additional fee.
2
Record the delayed cash outlay.
$C_{save} = AUD 1560$
The supplied later purchase price is paid entirely from savings.
3
Calculate the difference.
$C_{borrow} - C_{save} = 1650 - 1560 = AUD 90$
The saving-first purchase uses AUD 90 less in the stated purchase-outlay comparison.
4
Interpret the comparison's scope.
The totals alone do not value earlier use of the item or any financial effects omitted from the question.

Answer

Borrowing costs AUD 1650; saving-first purchase costs AUD 1560, AUD 90 less.

03 / YOUR TURN

Try it yourself.

Check your understanding

A AUD 2000 balance incurs 1% interest and a AUD 150 end-of-month payment. Find the new balance.

Show answer & reasoning

AUD 1,870.00

Interest is $2000 (0.01) = AUD 20$ . Add it before the payment: $B_{next} = 2000 + 20 - 150 = AUD 1870.00$ . The principal reduces by AUD 130 because AUD 20 of the payment covers interest.

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Loan balance AUD 5000, monthly rate 0.6%, month-end payment AUD 250. Find the new balance.
2 marks
A AUD 10000 loan incurs 0.5% monthly interest. What is the interest in its first month?
2 marks
A AUD 10000 balance has 0.5% monthly interest and only AUD 40 repaid at month end. Does it fall? Find the new balance.
2 marks
Find the balance after 3 monthly periods on AUD 2000 at 1% per month with AUD 300 month-end payments.
2 marks
Cash price AUD 2200. A loan requires 24 payments of AUD 102 plus AUD 50 fee. Find the extra cost.
2 marks
A comparison assumes a constant loan rate and unchanged future cash price. Give two reasons its actual outcome could differ.
2 marks

Read the worked solutions

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

Loan balance AUD 5000, monthly rate 0.6%, month-end payment AUD 250. Find the new balance.
2 marks
$5000 \times 1.006 - 250 = 4780$ .
AUD 4,780.00
A AUD 10000 loan incurs 0.5% monthly interest. What is the interest in its first month?
2 marks
$10000 \times 0.005 = 50$ .
AUD 50.00
A AUD 10000 balance has 0.5% monthly interest and only AUD 40 repaid at month end. Does it fall? Find the new balance.
2 marks
$Interest = 50$ .
$10000 + 50 - 40 = 10010$ .
No; it rises to AUD 10010.
Find the balance after 3 monthly periods on AUD 2000 at 1% per month with AUD 300 month-end payments.
2 marks
$B_{1} = 1720$ .
$B_{2} = 1437.20$ .
$B_{3} = 1437.20 \times 1.01 - 300 = 1151.572$ .
AUD 1,151.57
Cash price AUD 2200. A loan requires 24 payments of AUD 102 plus AUD 50 fee. Find the extra cost.
2 marks
$24 \times 102 + 50 = 2498$ .
$2498 - 2200 = 298$ .
AUD 298.00
A comparison assumes a constant loan rate and unchanged future cash price. Give two reasons its actual outcome could differ.
2 marks
These variables are assumptions rather than guaranteed facts.
The loan rate could change; the future cash price could change.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 General Mathematics: investing and borrowing. Supplied educational models cover interest, investment comparisons, shares and credit; they are not recommendations of real products or statements of personal tax liability.

Official curriculum source ↗

In this lesson

Understand the idea.

Track a reducing balance

Compare two decisions

See the method in action.

Worked example

Calculate the first month's interest.

Subtract the first end-of-month payment.

Calculate the second month's interest on the new balance.

Subtract the second payment.

Worked example

Find the full borrowing outlay.

Record the delayed cash outlay.

Calculate the difference.

Interpret the comparison's scope.

Try it yourself.

Check your understanding

Keep practising