IB Maths AA HLTopic 1 — Number & AlgebraPaper 1 & 2~10 min read
Compound Interest & Depreciation
This is the financial maths note. Compound interest is when interest is added to both the original amount and the interest already accumulated — so growth speeds up over time. Depreciation is the same idea in reverse: a value that loses a fixed percentage each period, like a car or a laptop. Both follow geometric-style formulas, with one twist (the compounding-periods variable k) that catches students out.
📘 What you need to know
Simple interest adds the same fixed amount each year (arithmetic). Compound interest adds interest on top of interest (geometric).
Compound interest formula: FV = PV · (1 + r100k)kn✓ in formula booklet.
k is the number of compounding periods per year: 1 = annually, 2 = half-yearly, 4 = quarterly, 12 = monthly.
More frequent compounding gives slightly more interest for the same nominal rate. Monthly > quarterly > annual.
Depreciation formula: FV = PV · (1 − r100)n✗ NOT in formula booklet — but you can derive it from the compound interest formula by setting k = 1 and changing + to −.
“Find when an investment doubles” or “find when a car halves” → set up an equation and solve using logarithms.
Simple vs compound interest
The two interest types behave very differently over long time periods. Understanding the difference is the conceptual heart of this topic.
📏
Simple interest
Same fixed amount of interest each year — based only on the original amount.
FV = PV(1 + rn100)
Linear growth → arithmetic in flavour
📈
Compound interest
Interest grows on interest — the balance at year-end becomes the new “principal” for next year.
FV = PV(1 + r100)n
Exponential growth → geometric in flavour
$1000 invested at 8%/year — simple vs compound (10 years)
After 10 years on $1000 at 8%, simple interest gives you $1800 (a flat +$80 each year). Compound interest gives you $2159 — the gap of nearly $360 is “interest on interest”. Over 30+ years the compound curve absolutely dominates. This is why early saving matters.
Compounding periods — the k variable
Banks don’t always wait a full year before adding interest. Many add it monthly, quarterly, or even daily. The nominal annual rate stays the same, but it gets divided into smaller chunks and applied more frequently.
Frequency
k
Periods/year
Rate per period
Annually
1
once per year
r%
Half-yearly
2
every 6 months
r/2 %
Quarterly
4
every 3 months
r/4 %
Monthly
12
every month
r/12 %
Example: 6% per annum compounded monthly means 0.5% interest is added each month — and there are 12 of these per year, so 60 of them in 5 years.
🤔 Why does monthly compounding give more than annual?
Because each month’s interest is added to the balance, and then next month’s interest is calculated on the new (slightly larger) balance. With annual compounding, you wait the full 12 months before getting any growth-on-growth boost. The more compounding periods, the more “interest on interest” gets squeezed in.
The compound interest formula
Compound interest
FV = PV · (1 + r100k)kn✓ in formula booklet
What each piece means: FV = future value (what you end up with) | PV = present value (your starting deposit) r% = nominal annual interest rate | k = compounding periods per year n = number of years (not periods!)
Watch the units carefully. n is in years, not periods. The formula handles the conversion automatically — kn in the exponent gives you the total number of compounding events.
🧭 Recipe — compound interest question
Identify each variable: starting amount (PV), nominal rate r, compounding frequency k, number of years n.
Substitute into the formula. Don’t simplify r/100k in your head — leave it for the calculator.
Use your GDC. The formula booklet’s compound interest formula is also accessible through most graphing calculators’ Finance package.
Round at the end, to whatever precision the question asks for.
Compound depreciation
Depreciation is the same idea as compound interest, but the value decreases by a fixed percentage each year instead of increasing. Cars, laptops, and machinery typically follow this pattern in the early years of their lifespan.
Compound depreciation (annual)
FV = PV · (1 − r100)n✗ NOT in formula booklet — but easy to derive
🤔 How to derive it from the booklet
Take the compound interest formula, set k = 1 (depreciation almost always compounds annually), and replace the + with a − because the value is shrinking instead of growing. That gives you FV = PV(1 − r/100)n. If you ever forget the depreciation formula in the exam, it’s a one-line reconstruction from the compound interest formula in your booklet.
GDC tip: if you’re using your calculator’s Finance package for a depreciation question, just enter the interest rate as a negative number (e.g. −15 instead of 15). The same formula handles both growth and decay this way.
Worked examples
WE 1
Compound interest — compounded annually
Maria deposits $5000 into an account that pays 4% per year compounded annually. Find the value of the account after 8 years, to the nearest dollar.
Step 1: Identify variablesPV = 5000, r = 4, k = 1, n = 8Step 2: Substitute into formulaFV = 5000 × (1 + 4100 × 1)1 × 8= 5000 × (1.04)8Step 3: Compute on GDC= 5000 × 1.36857…= 6842.85…≈ $6843
WE 2
Compound interest — compounded monthly
Daniel invests $3000 in an account paying a nominal annual interest rate of 6% compounded monthly. How much will be in the account after 4 years, to the nearest cent?
Step 1: Identify variablesPV = 3000, r = 6, k = 12, n = 4Step 2: SubstituteFV = 3000 × (1 + 6100 × 12)12 × 4= 3000 × (1.005)48Step 3: Compute on GDC= 3000 × 1.27049…= 3811.47…≈ $3811.47k = 12 means 0.5% added every month, 48 times total over 4 years
WE 3
Compare compounding frequencies
Two banks both offer a nominal annual rate of 4%. Bank A compounds monthly; Bank B compounds annually. If $10,000 is invested in each for 5 years, how much more does Bank A return?
Step 1: Bank A — monthly compoundingFVA = 10000 × (1 + 41200)60= 10000 × 1.22099… = $12,209.97Step 2: Bank B — annual compoundingFVB = 10000 × (1.04)5= 10000 × 1.21665… = $12,166.53Step 3: Find the differenceFVA − FVB = 12209.97 − 12166.53Bank A returns $43.44 moresmall but real — and grows larger for higher rates and longer periods
WE 4
Depreciation — value after n years
A new laptop costs $1500. It depreciates by 12% each year. Find its value after 6 years, to the nearest dollar.
A piece of machinery is bought for $25,000 and depreciates at 18% per year. Find the time, in years and months, for the value to fall to $12,500 (half its original value).
Step 1: Set up the equation12500 = 25000 × (0.82)n0.5 = (0.82)nStep 2: Take logs of both sidesln(0.5) = n · ln(0.82)n = ln(0.5)ln(0.82)= −0.6931−0.1985 = 3.491…Step 3: Convert decimal years → years and months3 years + 0.491 × 12 months= 3 years + 5.89… months≈ 3 years 6 monthsalways state both signs of the logs are negative — they cancel and give a positive n
💡 Top tips
Identify k first. The wording usually tells you (“compounded monthly” → 12, “compounded quarterly” → 4). Annual is the default if not stated.
The exponent is kn, not n. Forgetting to multiply k by n is the most frequent slip on this topic.
Keep PV positive in the compound interest formula. Don’t try to “encode” depreciation by making PV negative — use the depreciation formula instead, or flip the sign on r.
For “find the time” questions, take natural logs of both sides and use the power rule.
Years-and-months conversions: if n = 3.491, then it’s 3 years + 0.491 × 12 ≈ 5.89 months → 3 years 6 months.
The compound interest formula is in the booklet. The depreciation formula is not — but you can derive it. Either way, write it down before substituting.
For Paper 2 (calculator), use your GDC’s Finance package. For Paper 1, use the formula and your algebra skills.
⚠ Common mistakes
Using n in the exponent instead of kn. If k = 12 and n = 5, the exponent is 60, not 5.
Confusing nominal rate with effective rate. “6% per annum compounded monthly” means a yearly nominal rate of 6%, broken into monthly chunks of 0.5% — not 6% per month.
Forgetting that depreciation isn’t in the booklet. Memorise it, or derive it from compound interest.
Treating depreciation as simple subtraction. 15% depreciation per year on $1000 doesn’t take $150 off each year — it takes 15% off the current value, which shrinks over time.
Mixing simple and compound formulas. Re-read the question — “simple interest” and “compound interest” require different formulas.
Rounding mid-calculation. For Paper 2, keep full precision until the final answer.
Reading the wrong unit for time. If the question gives months but the formula expects years, convert. If a problem says “after 30 months” with monthly compounding, that’s n = 2.5 years and kn = 30.
Compound interest is a beautiful demonstration of geometric growth at work — every formula in this note is just u1 · rn in disguise. The “growth factor” (1 + r/100k) is the common ratio; kn is the number of multiplications. If you ever forget a formula, fall back on geometric reasoning.
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