IB Maths
Paper 1 & 2
15 min read
Compound Interest & Depreciation
Money grows (interest) or shrinks (depreciation) by a fixed percentage each period. One simple formula handles both. Let’s go.
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What you need to know
- What compound interest means and how it differs from simple interest
- The compound interest formula with compounding periods
- How to handle depreciation (just compound interest going down)
- How to find missing values: future value, time, or interest rate
Two simple ideas: growth and decay
Compound interest (going up ↑)
Money you save earns a % each year, and next year you earn interest on the bigger amount too.
Examples:
- Bank savings
- Investments
- Inflation
Depreciation (going down ↓)
The value of something drops by a % each year (cars, electronics, machinery).
Examples:
- Car value over time
- Mobile phone resale
- Computer / laptop
Both work the same way. The only difference: interest adds a percentage, depreciation subtracts a percentage.
Formula for compound interest
This is the formula in your IB formula booklet:
FV = PV × (1 + r100k)kn
What each letter means
- FV = Future Value (what you’ll have at the end)
- PV = Present Value (what you start with)
- r = annual interest rate (as a percentage, e.g. 5)
- k = compounding periods per year (1, 2, 4, or 12)
- n = number of years
What does k mean?
k tells you how often interest gets added each year:
- Yearly → k = 1 (interest added 1 time per year)
- Half-yearly → k = 2
- Quarterly → k = 4
- Monthly → k = 12
Important: the more often interest is compounded (bigger k), the more money you make over the same time.
Quick example: yearly compound interest
You invest $1000 at 5% per year for 3 years, compounded yearly.
- PV = 1000, r = 5, k = 1, n = 3
- FV = 1000 × (1 + 5100 × 1)1 × 3
- FV = 1000 × (1.05)3
- FV = 1000 × 1.157625 = $1157.63
Formula for depreciation
Same formula — just subtract instead of add, and use k = 1 (depreciation is usually yearly):
FV = PV × (1 − r100)n
Tip: this formula is NOT in the formula booklet — but you can derive it from the compound interest one by setting k = 1 and changing the + to a −.
Quick example: depreciation
A laptop worth $1200 loses 20% of its value each year. Find its value after 3 years.
- PV = 1200, r = 20, n = 3
- FV = 1200 × (1 − 20100)3
- FV = 1200 × (0.8)3
- FV = 1200 × 0.512 = $614.40
Worked Examples
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Example 1 — Yearly interest
Aisha invests $5000 at 4% annual interest, compounded yearly. How much will she have after 6 years?
Answer:
PV = 5000, r = 4, k = 1, n = 6
FV = 5000 × (1 + 4/100)6
= 5000 × (1.04)6
= 5000 × 1.26532…
FV ≈ $6326.60
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Example 2 — Compounded monthly
Kim invests MYR 2000 at 2.5% per year, compounded monthly. Find the amount after 5 years (to the nearest 10 MYR).
Answer:
Monthly compounding → k = 12.
PV = 2000, r = 2.5, k = 12, n = 5
FV = 2000 × (1 + 2.5/(100 × 12))12 × 5
= 2000 × (1 + 0.002083)60
= 2000 × 1.13300…
= 2266.00…
FV ≈ MYR 2270
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Example 3 — Find the value of a car (depreciation)
Kyle buys a new car for AUD $14 999. The car loses 15% of its value each year. Find its value after 5 years.
Answer:
PV = 14999, r = 15, n = 5
FV = 14999 × (1 − 15/100)5
= 14999 × (0.85)5
= 14999 × 0.4437…
FV ≈ AUD $6655
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Example 4 — Find how many years (using logs)
From Example 3, after how many years will the car’s value drop to about AUD $9999?
Answer:
9999 = 14999 × (0.85)n
Divide both sides:
9999 / 14999 = (0.85)n
0.6666… = (0.85)n
Take logs:
n = log(0.6666) / log(0.85)
n = 2.495…
2 full years + 0.495 × 12 ≈ 6 months
≈ 2 years 6 months
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Example 5 — Find the interest rate
$2000 grows to $2400 in 4 years compounded yearly. Find the annual interest rate r.
Answer:
2400 = 2000 × (1 + r/100)4
1.2 = (1 + r/100)4
Take 4th root:
1 + r/100 = (1.2)1/4
1 + r/100 = 1.04663…
r/100 = 0.04663…
r ≈ 4.66% per year
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Quick tips
- “Per annum” = “per year”. Always assume yearly unless told otherwise.
- For depreciation, just use (1 − r/100) instead of (1 + r/100).
- Use your GDC’s finance app as a backup. For depreciation, enter the rate as negative.
- To find n (number of years), you’ll usually need logarithms.
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Common mistakes
- Forgetting to multiply kn in the exponent. If interest is monthly for 5 years, the power is 12 × 5 = 60, not just 5.
- Putting r as a decimal in the formula. The formula uses r as a percentage (e.g. 5, not 0.05), because it’s already divided by 100 inside.
- Using “+” for depreciation. Depreciation = value drops, so always −.
- Confusing k with n. k = compounding periods per year. n = number of years. Don’t mix them up.
- Rounding too early. Keep many decimal places during calculation, only round at the end.
Final word: one formula does it all. Plug in your values, watch the signs (+ for growth, − for depreciation), and let your GDC do the heavy lifting. For “find n” or “find r” questions, use logs or the equation solver.
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