Solutions — Comparing Financial Options

Fluency $5 000 flat-rate loan at 8% p.a. over 2 years.

Interest = P × R × T = 5 000 × 0.08 × 2 = $800
Monthly repayment = (5 000 + 800) / 24 = 5 800 / 24 = $241.67

Fluency $5 000 reducing balance loan, $226.14/month for 24 months. Total interest paid.

Total paid = 226.14 × 24 = $5 427.36
Total interest = 5 427.36 − 5 000 = $427.36

Compare to the flat-rate loan above: the reducing balance loan at the same stated rate costs $800 − $427.36 = $372.64 less in interest.

Fluency $12 000 flat-rate loan at 10% p.a. over 3 years.

(a) Total interest = 12 000 × 0.10 × 3 = $3 600
(b) Monthly repayment = (12 000 + 3 600) / 36 = 15 600 / 36 = $433.33
(c) Total amount paid = $15 600

Fluency $10 000 deposit for 3 years: Account A (5% p.a. annual) vs Account B (5% p.a. monthly).

Account A (annual compounding):
A = 10 000 × 1.05³ = 10 000 × 1.157625 = $11 576.25

Account B (monthly compounding, r = 5/12 % = 0.4167%/month):
A = 10 000 × (1.004167)³⁶ = 10 000 × 1.16147 = $11 614.72

Account B gives $38.47 more. Monthly compounding earns interest on interest more frequently, producing a slightly higher effective annual rate of 5.116% vs the nominal 5%.

Understanding $20 000 car loan: flat rate 7% p.a. vs reducing balance 7% p.a., both over 4 years.

Option A (flat rate):
Interest = 20 000 × 0.07 × 4 = $5 600
Total cost = $25 600

Option B (reducing balance, $478.92 × 48 months):
Total cost = 478.92 × 48 = $22 988.16
Total interest = 22 988.16 − 20 000 = $2 988.16

Option B saves $2 611.84 in interest. The same stated rate of 7% costs dramatically more as a flat-rate loan because interest is charged on the full $20 000 for all 4 years, even after most of the principal has been repaid.

Understanding $15 000 over 5 years: Loan X (flat rate 6% p.a.) vs Loan Y (reducing balance, $290/month).

Loan X (flat rate):
Interest = 15 000 × 0.06 × 5 = $4 500
Monthly repayment = (15 000 + 4 500) / 60 = $325
Total cost = $19 500

Loan Y (reducing balance, $290 × 60 months):
Total cost = 290 × 60 = $17 400
Total interest = 17 400 − 15 000 = $2 400
Total cost = $17 400

Loan Y has the lower total cost by $19 500 − $17 400 = $2 100, despite lower monthly repayments than Loan X ($290 vs $325).

Understanding $30 000 loan, 9% p.a. (0.75% monthly). Standard $622.75/month vs $800/month.

Standard repayments ($622.75 × 60 months):
Total paid = $37 365 Total interest = $7 365

Increased repayments ($800/month):
Using PV formula to find n:
30 000 = 800 × (1 − 1.0075⁻ⁿ) / 0.0075
0.28125 = 1 − 1.0075⁻ⁿ
1.0075⁻ⁿ = 0.71875 → n ≈ 44.1 months

After 44 full payments the remaining balance ≈ $152 (paid in month 45).
Total paid ≈ 44 × $800 + $152 = $35 352
Total interest ≈ $35 352 − $30 000 = $5 352

Interest saved ≈ $7 365 − $5 352 = $2 013, and the loan is paid off 15 months sooner.

Understanding $50 000 for 5 years: Option A (4.2% p.a. annual) vs Option B (4.0% p.a. monthly).

Option A (4.2% compound annually):
A = 50 000 × 1.042⁵ = 50 000 × 1.2284 = $61 420

Option B (4.0% compound monthly, r = 0.3333%/month):
A = 50 000 × (1.003333)⁶⁰ = 50 000 × 1.2210 = $61 050

Option A gives $370 more. Even though Option B compounds more frequently (giving a higher effective rate than 4.0%), the higher nominal rate of Option A (4.2% vs 4.0%) more than compensates. Frequent compounding cannot overcome a 0.2% rate disadvantage over this timeframe.

Problem Solving $25 000 loan: flat rate 9% vs reducing balance 11% over 3 years.

(a) Total interest:
Option A (flat rate 9%):
Interest = 25 000 × 0.09 × 3 = $6 750
Monthly repayment = (25 000 + 6 750) / 36 = $881.94
Total cost = $31 750, total interest = $6 750

Option B (reducing balance 11%, $817.84 × 36 months):
Total cost = 817.84 × 36 = $29 442.24
Total interest = 29 442.24 − 25 000 = $4 442.24

(b) Option B is cheaper by $31 750 − $29 442.24 = $2 307.76, despite having a higher stated interest rate.

(c) Why the higher-rate option is cheaper:
A flat-rate loan charges interest on the original $25 000 for the entire 3 years — including months when the borrower has already repaid most of the principal. The effective rate is much higher than 9%. In contrast, the reducing balance loan charges 11% only on the outstanding balance, which shrinks each month. The interest bill falls progressively, making the actual cost significantly lower. The lesson: always compare total interest paid, not just the stated rate.

Problem Solving $450 000: Loan A (4.5%, 30 years, $2 280.08/month) vs Loan B (4.8%, 25 years, $2 567.24/month).

(a) Total interest:
Loan A: 2 280.08 × 360 months = $820 828.80 → interest = $820 828.80 − $450 000 = $370 828.80
Loan B: 2 567.24 × 300 months = $770 172 → interest = $770 172 − $450 000 = $320 172

(b) Why Loan B may be the better choice:
Despite higher monthly repayments ($287.16 more per month) and a higher stated rate (4.8% vs 4.5%), Loan B saves $50 657 in total interest over its lifetime. The 5-year shorter term eliminates 60 payments entirely. The compounding effect of interest over 30 years is so large that even a modest rate difference matters less than the number of years borrowing.

Over 5 fewer years at $2 280.08/month, the savings from not paying = 60 × $2 280.08 = $136 805 — far more than the extra $287/month paid for 25 years (25 × 12 × $287 = $86 100). Financially, shorter loan terms almost always win.