Solutions — Financial Applications

1. Identify the correct model. Fluency

   • (a) City population grows 3%/year: Exponential growth (compound). P = P₀ × (1.03)ⁿ. A constant percentage increase each period gives exponential growth.
   • (b) Photocopier loses $800/year: Straight-line depreciation. V = P − 800n. A fixed dollar loss each year is linear.
   • (c) Credit card at 18% p.a. monthly: Compound interest (multiple compounding periods). A = B(1 + 0.18/12)¹²ⁿ = B(1.015)¹²ⁿ.
   • (d) Bond pays $120/year on $2000: Simple interest. I = Prn: 120 = 2000 × r × 1 ⇒ r = 6% p.a. The same fixed amount each year is simple interest.

2. Loan calculations. Fluency

   • (a) $2000, 20% monthly, 1 year: A = 2000×(1 + 0.20/12)¹² = 2000×(1.01667)¹² ≈ 2000×1.21940 ≈ $2438.80.
   • (b) $8000, 12% monthly, 3 years: A = 8000×(1.01)³⁶ ≈ 8000×1.43077 ≈ $11446.
   • (c) $15000, 9% monthly, 5 years: A = 15000×(1.0075)⁶⁰ ≈ 15000×1.56568 ≈ $23485.
   • (d) $25000, 4.5% annual, 10 years: A = 25000×(1.045)¹⁰ ≈ 25000×1.55297 ≈ $38824.

3. Inflation calculations. Fluency

   • (a) House $400000, 3%, 5 years: C = 400000×(1.03)⁵ ≈ 400000×1.15927 ≈ $463708.
   • (b) Groceries $150, 4%, 10 years: C = 150×(1.04)¹⁰ ≈ 150×1.48024 ≈ $222.04.
   • (c) Car $25000 at 2.5% for 15 years: C = 25000×(1.025)¹⁵ ≈ 25000×1.44830 ≈ $36208.
   • (d) Prices double at 5%: n=14: 1.05¹⁴≈1.979 — not yet doubled. n=15: 1.05¹⁵≈2.079 — doubled! Prices double in approximately 15 years. (Rule of 72: 72÷5=14.4 years — close!)

4. Present value. Fluency

   • (a) $10000 in 5 years at 4%: P = 10000 ÷ (1.04)⁵ = 10000 ÷ 1.21665 ≈ $8219.
   • (b) $50000 in 10 years at 6%: P = 50000 ÷ (1.06)¹⁰ = 50000 ÷ 1.79085 ≈ $27920.
   • (c) $15000 in 3 years at 5%: P = 15000 ÷ (1.05)³ = 15000 ÷ 1.15763 ≈ $12958.
   • (d) Simple interest: $5000 in 4 years at 5%: A = P(1+rn) ⇒ 5000 = P×(1+0.05×4) = P×1.2 ⇒ P = 5000÷1.2 ≈ $4166.67.

5. Read from the graph: comparing three strategies. Understanding

   • (a) At year 4: X = 5000+500×4 = $7000. Y = 5000×1.08⁴ ≈ $6802. Z = 5000×1.06⁴ ≈ $6312. Strategy X has the most in year 4 (higher rate more than compensates for being simple vs compound at this stage).
   • (b) Y overtakes X: n=6: X=$8000, Y=5000×1.08⁶≈$7934 — X leads. n=7: X=$8500, Y=5000×1.08⁷≈$8569 — Y leads. Between year 6 and year 7.
   • (c) At year 10: X = 5000+5000 = $10000. Y = 5000×1.08¹⁰ ≈ $10795. Z = 5000×1.06¹⁰ ≈ $8954.
   • (d) Recommendation for 10 years: Strategy Y (compound 8%). It overtakes the simple 10% around year 7 and finishes $795 higher by year 10. For long-term investing, compound interest always wins — each year’s interest earns further interest, creating accelerating growth.

6. Savings goal. Understanding

   • (a) 8% for 5 years: 15000×(1.08)⁵ = 15000×1.46933 ≈ $22040. No — he will have $22040, which is less than $25000.
   • (b) Shortfall: 25000 − 22040 = $2960 short.
   • (c) Rate needed: (1+r)⁵ = 25000/15000 = 1.667 (= 5/3). Try r=10%: 1.1⁵=1.61051 ⇒ A=$24158 (not enough). Try r=11%: 1.11⁵=1.68506 ⇒ A=$25276 (just enough!). Rate needed ≈ 11% p.a.
   • (d) Principal for $25000 at 8%: P = 25000 ÷ (1.08)⁵ = 25000 ÷ 1.46933 ≈ $17016. He needs to invest $17016 today (not $15000) to guarantee $25000 after 5 years at 8%.

7. Credit card debt. Understanding

   • (a) Balance after 6 months: A = 3000×(1.02)⁶ ≈ 3000×1.12616 ≈ $3378. Over $378 added in just 6 months with no payments!
   • (b) Balance after 1 year: A = 3000×(1.02)¹² ≈ 3000×1.26824 ≈ $3805.
   • (c) Month-by-month with $100 payments: Month 1: 3000×1.02=$3060; after payment: $2960. Month 2: 2960×1.02=$3019.20; after payment: $2919.20. Month 3: 2919.20×1.02=$2977.58; after payment: $2877.58. Yes — the debt reduces by about $40 each month (payment of $100 minus ~$60 interest).
   • (d) Approximate months to clear: Interest per month ≈ $60; repayment = $100; net reduction = $40/month. 3000 ÷ 40 ≈ 75 months (≈ 6.25 years). In practice it’s slightly faster as the balance decreases.

8. Real estate vs shares. Understanding

   • (a) After 5 years: R = 200000×(1.05)⁵ ≈ $255256. S = 200000×(1.08)⁵ ≈ $293866.
   • (b) After 15 years: R = 200000×(1.05)¹⁵ ≈ $415786. S = 200000×(1.08)¹⁵ ≈ $634434.
   • (c) Difference after 15 years: 634434 − 415786 = $218648 more for shares. The 3% rate difference compounds dramatically over 15 years.
   • (d) Crash scenario: Year 5 value: $293866. After 30% crash: 293866×0.70 = $205706. Then 10 years at 8%: 205706×(1.08)¹⁰ ≈ 205706×2.15892 ≈ $444048. Even with a 30% crash in year 5, shares recover strongly to $444048 — still ahead of real estate ($415786).

9. Multi-phase investment. Problem Solving

   • (a) After 5 years at 7%: A₅ = 10000×(1.07)⁵ = 10000×1.40255 ≈ $14026.
   • (b) After 8 years total: A₈ = 14026×(1.09)³ = 14026×1.29503 ≈ $18163.
   • (c) Total interest: 18163 − 10000 = $8163 total interest earned over 8 years.
   • (d) Compare with 7% for all 8 years: 10000×(1.07)⁸ = 10000×1.71819 ≈ $17182. Switching to 9% earned an extra $981 (18163−17182). Moving to a higher rate in the later years amplified the gains on the accumulated amount.

10. Early loan repayment. Problem Solving

    • (a) No repayments, 3 years: A = 20000×(1.08)³ = 20000×1.25971 ≈ $25194.
    • (b) With $5000 repayment at end of year 1: End of year 1: 20000×1.08=$21600; after repayment=$16600. End of year 2: 16600×1.08=$17928. End of year 3: 17928×1.08=$19362.
    • (c) Saving from early payment: 25194 − 19362 = $5832 saved. The $5000 payment at year 1 reduced the final balance by $5832.
    • (d) Compare with investing $5000 at 8% for 2 years: 5000×(1.08)² = 5000×1.1664 = $5832. The two amounts are identical! This makes sense: whether you apply $5000 against debt at 8% or invest $5000 at 8%, the financial gain is exactly the same when the rates match. Paying off debt at rate r is equivalent to investing at rate r.