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Escape Room · Reducing Balance Loans
First Home
Coastal Heights Bank · Loan Approval System · Portal closes at 5pm
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Checks complete: 0 of 6 0%
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📍 Scene Setup Josh's parents are applying for a home loan at Coastal Heights Bank. The online approval system requires all repayment calculations to be manually verified before the portal accepts the application, and it closes at 5pm.

Josh has volunteered to run through the numbers. Six checks stand between his family and their first home.

📋 Loan Details, Coastal Heights Bank

Original loan (V₁)
$1000
Interest rate per period
10%
Regular repayment (d)
$300
Recurrence: Vₙ₊₁ = 1.1 × Vₙ − 300  ·  Interest charged = Vₙ × 0.10 each period
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Check 1, Period 1 Interest
The loan starts at $1000. How much interest is charged in the first period?
Interest = opening balance × interest rate. Here that's $1000 × 10%.
💡 Interest = $1000 × 0.10 = ?
✅ $100 interest charged. Now the repayment gets subtracted from that new balance.
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Check 2, Closing Balance Period 1
Opening balance was $1000. Interest of $100 was added. Then a $300 repayment was made.

What is the closing balance at the end of period 1?
Opening + interest − repayment = closing. Or use the recurrence: V₂ = 1.1 × 1000 − 300.
💡 V₂ = 1.1 × 1000 − 300 = 1100 − 300 = ?
✅ Closing balance = $800. The loan dropped by $200, not $300, because $100 of that was just covering interest.
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Check 3, Period 2 Interest
Period 2 opens with a balance of $800. How much interest is charged this period?
Same rate (10%), different balance. Interest = $800 × 0.10.
💡 $800 × 0.10 = ? Notice this is less than the $100 from period 1, the balance is reducing!
✅ $80 interest. Down from $100, the reducing balance is working.
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Check 4, Closing Balance Period 2
Period 2: opening $800, interest $80, repayment $300.

What is the closing balance at the end of period 2?
V₃ = 1.1 × 800 − 300 = ?
💡 800 + 80 − 300 = ? (or: 1.1 × 800 − 300)
✅ Closing balance = $580. Two periods down: $1000 → $800 → $580. Good progress.
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Check 5, Second Application
Another applicant has a loan of $500 with a 20% interest rate per period and a $200 repayment.

What is their closing balance after period 1?
Set up the recurrence: V₂ = R × V₁ − d. What is R for 20% interest?
💡 R = 1 + 0.20 = 1.2. V₂ = 1.2 × 500 − 200 = 600 − 200 = ?
✅ V₂ = $400. Higher interest rate means the balance reduces more slowly, same repayment, more going to interest.
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Check 6, Total Interest
Back to the main loan (V₁ = $1000, 10% per period, $300 repayment).

The interest charged in periods 1, 2 and 3 was $100, $80, and $58 respectively.

What is the total interest paid across these 3 periods?
Just add the three interest amounts together.
💡 $100 + $80 + $58 = ? (Period 3 opening was $580, so interest = $580 × 0.10 = $58)
✅ $238 total interest over 3 periods. The loan is down to $338, the verification is complete.
🏠
Loan Approved!
All 6 checks passed. The Coastal Heights Bank portal has accepted the application and issued a conditional approval. Josh's parents are one step closer to their first home.

$1000 → $800 → $580 → $338. $238 paid in interest over 3 periods. The maths checks out. 🏡
Vₙ₊₁ = R·Vₙ − d
R = 1 + r
Interest = Vₙ × r
Total interest = repaid − loan
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