What even is this? When you borrow money for a house, car, or phone plan, interest is usually calculated on the remaining balance, not the original loan. That's what makes it "reducing balance." Every repayment chips away at what you owe, which means the interest charge also shrinks over time. This is recurrence relations applied directly to real money.
If you only do one thing
Each period adds interest then subtracts the payment, so the balance shrinks to zero. Track it with the Ans key.
The 10-minute version
Low energy? Do just this and you have still won the day. Zero is the only fail.
The one move: read the first formula box and the first worked example above, then do Practice Question 1. Screenshot the win for Nat and stop there.
📺 Watch it explained
Four short ways in. The 🎬 cards are waiting on Nat's videos, and each has a ready-to-read film script tucked underneath.
Section 1 · How a Reducing Balance Loan Works
Interest on the remaining balance
Unlike flat-rate loans, the interest charged each period is calculated on whatever you still owe, not the original amount. So the interest gets smaller as you pay down the loan.
Each period: two things happen
First, interest is added to the outstanding balance. Then your repayment is subtracted. The order matters, interest is charged before you make your payment.
1
Add interest
Balance × interest rate
Balance × interest rate
→
2
Subtract repayment
New balance = result − d
New balance = result − d
→
3
Repeat
This is the next period's opening balance
This is the next period's opening balance
Section 2 · The Recurrence Relation
A reducing balance loan is just a recurrence relation where the multiplier (r) includes the interest and d is the repayment amount being subtracted.
An+1 = r × An − d
Aₙ = outstanding balance at period n
r = 1 + i = interest multiplier (e.g. 10% → r = 1.1)
d = regular repayment amount (positive)
A₁ = original loan amount (always given)
⚠️ Easy to mix up: the interest rate i goes inside r = 1 + i. A 5% interest rate means r = 1.05. And d is subtracted (it's a repayment), so don't forget the minus sign in the recurrence. If you write + d by mistake, you'll be adding to the loan instead of paying it off.
Section 3 · Building a Repayment Schedule
A repayment schedule shows exactly what happens each period. The exam often asks you to complete or interpret one of these tables.
| Period | Opening balance | Interest (×10%) | Repayment | Closing balance |
|---|---|---|---|---|
| 1 | $1000.00 | +$100.00 | −$300.00 | $800.00 |
| 2 | $800.00 | +$80.00 | −$300.00 | $580.00 |
| 3 | $580.00 | +$58.00 | −$300.00 | $338.00 |
| 4 | $338.00 | +$33.80 | −$300.00 | $71.80 |
Notice how the interest amount decreases each period, $100, then $80, then $58, then $33.80. That's the "reducing balance" in action. The interest is always recalculated on the new (smaller) balance.
Section 4 · Calculating Total Interest Paid
💡 Key formula, total interest
Total interest = Total repaid − Original loan
Add up all the repayments (including the final smaller one if it exists). Subtract the original loan amount. What's left is the interest you paid, the cost of borrowing.
Example: If you borrowed $1000 and made 4 full repayments of $300 plus a final payment of $71.80, total repaid = $1271.80. Interest = $1271.80 − $1000 = $271.80.
Example: If you borrowed $1000 and made 4 full repayments of $300 plus a final payment of $71.80, total repaid = $1271.80. Interest = $1271.80 − $1000 = $271.80.
Section 5 · Worked Example
🏠 The Coastal Heights Purchase
Josh's parents take out a loan of $1000 (simplified!) at 10% interest per period, with a regular repayment of $300 per period. They want to track the loan balance and understand the total cost of borrowing.
Step 1 · Write the recurrence relation
Set up the formula with the given values.
Original loan: A₁ = $1000
Interest rate: i = 10% → r = 1 + 0.10 = 1.1
Repayment: d = $300
Aₙ₊₁ = 1.1 × Aₙ − 300, A₁ = 1000
Interest rate: i = 10% → r = 1 + 0.10 = 1.1
Repayment: d = $300
Aₙ₊₁ = 1.1 × Aₙ − 300, A₁ = 1000
Step 2 · Generate the first 4 terms
What is the outstanding balance after each repayment period?
A₂ = 1.1 × 1000 − 300 = 1100 − 300 = $800
A₃ = 1.1 × 800 − 300 = 880 − 300 = $580
A₄ = 1.1 × 580 − 300 = 638 − 300 = $338
A₅ = 1.1 × 338 − 300 = 371.80 − 300 = $71.80
A₆ = 1.1 × 71.80 − 300 = 78.98 − 300 < 0 → loan paid off in period 5 with a smaller final payment
A₃ = 1.1 × 800 − 300 = 880 − 300 = $580
A₄ = 1.1 × 580 − 300 = 638 − 300 = $338
A₅ = 1.1 × 338 − 300 = 371.80 − 300 = $71.80
A₆ = 1.1 × 71.80 − 300 = 78.98 − 300 < 0 → loan paid off in period 5 with a smaller final payment
Step 3 · Total interest over the first 3 periods
How much interest was charged in periods 1, 2 and 3 combined?
Period 1 interest: 1000 × 0.10 = $100
Period 2 interest: 800 × 0.10 = $80
Period 3 interest: 580 × 0.10 = $58
Total = 100 + 80 + 58 = $238
Period 2 interest: 800 × 0.10 = $80
Period 3 interest: 580 × 0.10 = $58
Total = 100 + 80 + 58 = $238
Step 4 · Interpret
Why is the repayment amount important?
Each period, the repayment of $300 exceeds the interest charged, so the balance is genuinely reducing. If the repayment were less than the interest (e.g. only $50), the balance would grow each period and the loan would never be paid off. This is how people get trapped in debt spirals.
Section 6 · Practice Questions
Tap to reveal the worked answer.
Question 1
A loan has A₁ = $500, interest rate 20% per period, repayment $200 per period.
Write the recurrence relation and find A₂.
Write the recurrence relation and find A₂.
▼
r = 1 + 0.20 = 1.2 · d = 200
Recurrence: Aₙ₊₁ = 1.2 × Aₙ − 200, A₁ = 500
A₂ = 1.2 × 500 − 200 = 600 − 200 = $400
Recurrence: Aₙ₊₁ = 1.2 × Aₙ − 200, A₁ = 500
A₂ = 1.2 × 500 − 200 = 600 − 200 = $400
✓ Aₙ₊₁ = 1.2Aₙ − 200, A₂ = $400
Question 2
Continue from Q1 (A₂ = $400). Find A₃.
▼
A₃ = 1.2 × A₂ − 200
A₃ = 1.2 × 400 − 200
A₃ = 480 − 200 = $280
A₃ = 1.2 × 400 − 200
A₃ = 480 − 200 = $280
✓ A₃ = $280
Question 3
A loan of $3000 charges 15% per period and requires a repayment of $500.
Write the recurrence and find the balance after the first repayment (A₂).
Write the recurrence and find the balance after the first repayment (A₂).
▼
r = 1.15 · d = 500
Recurrence: Aₙ₊₁ = 1.15Aₙ − 500, A₁ = 3000
A₂ = 1.15 × 3000 − 500 = 3450 − 500 = $2950
Recurrence: Aₙ₊₁ = 1.15Aₙ − 500, A₁ = 3000
A₂ = 1.15 × 3000 − 500 = 3450 − 500 = $2950
✓ Aₙ₊₁ = 1.15Aₙ − 500, A₂ = $2950
Question 4, Repayment schedule
Complete this table for: A₁ = $800, r = 1.1, d = $150.
Period 1: Opening $800, Interest ?, Repayment $150, Closing ?
Period 2: Opening ?, Interest ?, Repayment $150, Closing ?
Period 1: Opening $800, Interest ?, Repayment $150, Closing ?
Period 2: Opening ?, Interest ?, Repayment $150, Closing ?
▼
Period 1: Interest = 800 × 0.1 = $80. Closing = 800 + 80 − 150 = $730
Period 2: Interest = 730 × 0.1 = $73. Closing = 730 + 73 − 150 = $653
Period 2: Interest = 730 × 0.1 = $73. Closing = 730 + 73 − 150 = $653
✓ Period 1 closing: $730 · Period 2 closing: $653
Question 5, Total interest
Using the original worked example (A₁ = $1000, 10% interest, $300 repayment), the balance after 3 repayments is $338.
How much total interest was paid across the first 3 periods?
How much total interest was paid across the first 3 periods?
▼
Method 1, add up each period's interest:
Period 1: $100 · Period 2: $80 · Period 3: $58
Total = 100 + 80 + 58 = $238
Method 2, total repaid minus reduction in loan:
Total repaid = 3 × $300 = $900
Loan reduction = $1000 − $338 = $662
Interest = $900 − $662 = $238 ✓ (same answer either way)
Period 1: $100 · Period 2: $80 · Period 3: $58
Total = 100 + 80 + 58 = $238
Method 2, total repaid minus reduction in loan:
Total repaid = 3 × $300 = $900
Loan reduction = $1000 − $338 = $662
Interest = $900 − $662 = $238 ✓ (same answer either way)
✓ Total interest = $238
First Home, Escape Room
The bank's loan approval system needs Josh's calculations verified before it processes the mortgage. 6 challenges on reducing balance loans, interest, balances, and total cost.
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