Compound Interest • Topic 3 of 4

Growth & Depreciation

Compound interest is just the money version of compound growth, so the same formula models populations, production, prices and bacteria, while flipping the sign models depreciation — the steady percentage loss in the value of machinery, vehicles or equipment. For a quantity rising r% per period, value = P(1 + r/100)ⁿ; for one falling r% per period, value = P(1 − r/100)ⁿ. The CAT favourite is mixed-sign growth: a population that grows in some years and shrinks in others, where you simply chain the multipliers (e.g. 1.1 × 0.9 = 0.99, a net 1% fall). For "find the value n years ago", divide instead of multiply: original = current ÷ (1 ± r/100)ⁿ. Note that depreciation here is the reducing-balance (written-down-value) method — each year’s fall is a percentage of the latest value, not of the original cost.

✅ Solved examples

1. A machine costing ₹50,000 depreciates 10% per annum. Find its value after 2 years.
Value = 50000 × 0.9 × 0.9 = 50000 × 0.81 = ₹40,500.
2. A town’s population is 64,000 and grows 25% per year. Find the population after 2 years.
25% ⇒ ×5/4. 64000 × 5/4 × 5/4 = 64000 × 25/16 = 100,000.
3. A car bought for ₹8,00,000 depreciates 20% in the first year and 10% in the second. Find its value after 2 years.
Value = 800000 × 0.8 × 0.9 = 800000 × 0.72 = ₹5,76,000.
4. A population grew 10% in one year then fell 10% the next. Find the net percentage change.
Multiplier = 1.1 × 0.9 = 0.99 ⇒ a net decrease of 1%.

✏️ Practice — try these, take hints as needed

1. A machine worth ₹1,00,000 depreciates 20% per annum. Value after 2 years?
Multiply by 0.8 each year.
100000 × 0.8 × 0.8.
= 100000 × 0.64.
₹64,000
2. A village of 10,000 grows 10% per year. Population after 3 years?
Multiply by 1.1 thrice.
10000 × 1.331.
Round to whole people.
13,310
3. An asset depreciates 10% then 20% in two years to reach ₹72,000. Find its original value.
Work backwards: divide.
72000 ÷ (0.9 × 0.8).
72000 ÷ 0.72.
₹1,00,000
4. A city’s population fell 20% then rose 25%. Net percentage change?
Chain the multipliers.
0.8 × 1.25.
= 1.00.
No change (0%)
5. Bacteria triple every hour. If there are 200 now, how many after 3 hours?
Multiplier is 3 each hour.
200 × 3³.
= 200 × 27.
5,400

📝 Topic test — 8 questions

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