Reducing-balance loans, 4 ways

What it is

A reducing-balance loan is paid off with fixed repayments. Each period two things happen in order: interest is added to what you still owe, then your repayment is taken off. The balance falls a little each period until it hits zero.

The one move, every period

new balance = old × (1 + i) − repayment

i = interest rate per period (as a decimal) · repayment = the fixed amount you pay each period.

Monthly loan? Find the rate per period first: annual rate ÷ 12. (12% per year compounding monthly = 1% per month = 0.01.)

Worked example

$10,000 loan at 12% p.a. compounding monthly (so 1% per month), repaying $300 a month.

Rate per period: 12% ÷ 12 = 1% = 0.01, so the multiplier is 1.01
Month 1: 10000 × 1.01 = 10100, then − 300 = $9,800.00
Month 2: 9800 × 1.01 = 9898, then − 300 = $9,598.00
Month 3: 9598 × 1.01 = 9693.98, then − 300 = $9,393.98

How a marker wants it laid out

Same question, written the QCAA way. Each line earns its own tick, so you score even if the final number slips.

Monthly rate: i = 12% ÷ 12 = 0.01 ✓ finds the rate

A_n+1 = 1.01 A_n − 300 ✓ writes the recurrence

A₀ = 10 000
A₁ = 1.01 × 10 000 − 300 = 9800
A₂ = 1.01 × 9800 − 300 = 9598
A₃ = 1.01 × 9598 − 300 = 9393.98 ✓ shows the iteration

After 3 months the balance is $9,393.98. ✓ answer in context

The four moves every time: state the rate, write the rule, show the steps, answer in a sentence with the $ and units.

Month by month

End of month	Calculation	Balance
Start	opening	$10,000.00
1	10000 × 1.01 − 300	$9,800.00
2	9800 × 1.01 − 300	$9,598.00
3	9598 × 1.01 − 300	$9,393.98

Notice the interest part shrinks each month (it is charged on a smaller balance), so more of every $300 goes to the actual loan.

Watch out

• Interest goes on first, then you subtract the repayment. Not the other way round.
• Use the rate per period (divide the yearly rate by how many times it compounds).
• The repayment is a dollar amount you subtract, never a number you multiply by.

The move, colour coded

new = old × (1 + i) − repayment

old = what you still owe (1 + i) = adds the interest − repayment = your payment comes off

One step per month

$10,000×1.01 −300 → $9,800×1.01 −300 → $9,598×1.01 −300 → $9,393.98

Grow by the interest, take off the payment. Repeat.

The balance heads to zero

Each repayment is bigger than the interest charged, so the balance drops, faster and faster, until it reaches zero.

Warm up first

Don't read yet, just have a go in your head:

Which happens first each period: add interest, or subtract the repayment?

Add the interest first (× the multiplier), then subtract the repayment.

Loan at 6% per year, compounding monthly. Rate per month?

6% ÷ 12 = 0.5% per month, so you multiply by 1.005.

Faded example: $5,000 at 2% per period, repaying $1,000

Rung 1 · watch one done fully

5000 × 1.02 = 5100, − 1000 = $4,100 → 4100 × 1.02 = 4182, − 1000 = $3,182

Rung 2 · you fill the gaps

5000 × 1.02 − 1000 = ? → next period − 1000 = ?

Check my gaps

$4,100, then $3,182.

Rung 3 · all you

$8,000 at 1% per period, repaying $2,000. Find the balance after 2 periods, then check.

Check my answer

8000 × 1.01 − 2000 = $6,080, then 6080 × 1.01 − 2000 = $4,140.80.

Say it back

In one sentence, out loud: what are the two things that happen to the balance each period, and in what order? If you can say it, you've got it.

⚡ Reducing-balance loans, one look

Each periodnew = old × (1 + i) − repayment

Orderinterest on first, then take the payment off

Per periodyearly rate ÷ times it compounds (12% monthly → 1%)

Example$10,000 @1%/mo, repay $300 → after 1 mo $9,800

Trapinterest before repayment · subtract the payment, don't multiply

On the Casio10000= then Ans×1.01−300= = =

The Ans key trick: type the loan, press =, then build the rule once and keep pressing = to step down the table.