Compound Interest and Exponential Growth

Why interest snowballs

Save money in a bank and it earns interest. With simple interest you earn the same amount each year, calculated only on your original deposit. With compound interest you earn interest on your interest too — so each year's gain is bigger than the last. Over time this difference becomes enormous, and the same idea drives population growth, viruses and debt.

This lesson builds on percentages — if finding a percentage of an amount is rusty, review percentage of an amount first.

The multiplier method

The fastest way to handle repeated percentage change is the multiplier.

• A p% increase means you keep 100% and add p%, so multiply by (1 + p/100).
• A p% decrease means you keep (100 − p)%, so multiply by (1 − p/100).

Change	Multiplier
+5%	1.05
+12%	1.12
−10%	0.90
−15%	0.85
+100%	2.00

Worked example 1: Increase £200 by 8%.

1. Multiplier for +8% is 1.08.
2. 200 × 1.08 = £216.

The real power appears when the change repeats. Each period you multiply by the same factor again.

The compound interest formula

If an amount P changes by r% each period for n periods, the final amount is:

final amount = P × (1 + r/100)ⁿ

The power n is what makes it grow so fast — multiplying by the multiplier over and over.

Worked example 2: £500 is invested at 4% compound interest per year for 3 years. Find the final amount.

1. P = 500, r = 4, n = 3. Multiplier = 1.04.
2. final = 500 × 1.04³.
3. 1.04³ = 1.04 × 1.04 × 1.04 = 1.124864.
4. final = 500 × 1.124864 = £562.43 (to the nearest penny).

Worked example 3: How much interest is earned if £1000 is invested at 5% compound for 4 years?

1. final = 1000 × 1.05⁴.
2. 1.05⁴ = 1.21550625.
3. final = £1215.51.
4. Interest = 1215.51 − 1000 = £215.51.

Compare with simple interest: 5% of £1000 is £50 per year × 4 = £200. Compounding earned £15.51 more, and the gap widens every extra year.

Comparing simple and compound year by year

For £1000 at 10% per year:

Year	Simple interest balance	Compound interest balance
0	£1000	£1000
1	£1100	£1100
2	£1200	£1210
3	£1300	£1331
4	£1400	£1464.10

Simple interest adds a flat £100 each year (a straight line). Compound interest adds more each year because 10% is taken of a bigger balance — that is exponential growth.

Exponential decay

If the multiplier is less than 1, the amount shrinks each period — exponential decay. The most common use is depreciation, where things like cars and phones lose a percentage of value yearly.

Worked example 4: A car worth £8000 loses 15% of its value each year. Find its value after 3 years.

1. Multiplier for −15% is 0.85.
2. value = 8000 × 0.85³.
3. 0.85³ = 0.614125.
4. value = 8000 × 0.614125 = £4913 (to the nearest pound).

Worked example 5: A population of bacteria grows by 20% every hour, starting at 500. Find the population after 5 hours.

1. Multiplier = 1.20, n = 5.
2. 500 × 1.2⁵ = 500 × 2.48832 = 1244.16.
3. About 1244 bacteria (numbers of organisms are whole, so round sensibly).

Different compounding periods

Interest is not always yearly. If a rate is given per year but applied monthly, divide the rate by 12 and count months.

Worked example 6: £2000 is invested at 6% per year, compounded monthly, for 2 years.

1. Monthly rate = 6 ÷ 12 = 0.5%, so multiplier = 1.005.
2. Number of months n = 2 × 12 = 24.
3. final = 2000 × 1.005²⁴.
4. 1.005²⁴ ≈ 1.12716, so final ≈ £2254.32.

More frequent compounding earns slightly more than yearly compounding at the same headline rate.

Doubling time and the Rule of 72

A handy estimate: under growth of r% per period, the amount roughly doubles after 72 ÷ r periods. This is the Rule of 72.

Worked example 7: At 8% per year, roughly how long until savings double?

1. 72 ÷ 8 = 9 years (approximately).

Why compounding beats simple interest: Simple interest always works on the original amount, so each year's gain is identical. Compounding works on the current balance, which keeps growing — so you are taking a percentage of a larger and larger number. Growth feeding on itself is the definition of exponential, and it is why starting to save early matters so much.

Where this is used

Compound interest decides how savings, ISAs, mortgages and credit-card debt grow. Exponential growth models populations, viral videos and the early spread of diseases. Exponential decay models depreciation of assets, radioactive material and medicine leaving the bloodstream. Understanding multipliers lets you compare loans, choose savings accounts, and judge whether "20% growth a month" is sustainable (usually not for long).

Practice activity

1. Increase £350 by 12% using a multiplier.
2. £600 is invested at 3% compound interest for 2 years. Find the final amount.
3. A laptop worth £900 depreciates 20% per year. Find its value after 2 years.
4. £1500 at 5% compound for 3 years — how much interest is earned?
5. Use the Rule of 72 to estimate the doubling time at 6% per year.

Answers:

1. 350 × 1.12 = £392.
2. 600 × 1.03² = 600 × 1.0609 = £636.54.
3. 900 × 0.8² = 900 × 0.64 = £576.
4. 1500 × 1.05³ = 1500 × 1.157625 = £1736.44; interest = £236.44.
5. 72 ÷ 6 = 12 years (approximately).

Summary

Compound interest earns interest on previous interest, growing exponentially. Use the multiplier: (1 + p/100) for an increase, (1 − p/100) for a decrease. The compound formula is final = P × (1 + r/100)ⁿ, where the power n drives the rapid growth. A multiplier above 1 is growth; below 1 is decay (depreciation). For non-annual periods, scale the rate and the count to match. The Rule of 72 estimates doubling time as 72 ÷ r.