Compound interest step by step

TL;DR:

• Compound interest earns interest on past interest, so each year's growth is a bigger amount.

• After n years, A = P(1 + r)ⁿ. For $1,000 at 5% for 3 years gives 1000 × 1.05³ ≈ 1157.63.

• Simple interest only grows by the same dollar amount each year.

You put $1,000 in an account that pays 5% a year. After one year you have $1,050. What happens in year two? If the bank adds interest to the original $1,000 only, that is simple interest. If it adds interest to the whole $1,050, that is compound interest. The difference looks tiny at first and gets larger over time.

Growth by a factor

Increasing by 5% is the same as multiplying by 1.05, since 1 + 0.05 = 1.05. One year of growth, then, is one multiplication. Three years of growth is three multiplications, which is the idea behind the formula A = P(1 + r)ⁿ, where P is the starting amount, r is the yearly rate written as a decimal, n is the number of years and A is the final amount.

Worked example: $1,000 growing for 3 years at a rate of 0.05

1. Year 1: 1000 × 1.05 = 1050.

2. Year 2: 1050 × 1.05 = 1102.50. The interest this year is 1102.50 − 1050 = 52.50, more than the first year's 50.

3. Year 3: 1102.50 × 1.05 = 1157.625, which rounds to $1,157.63.

The formula agrees: 1.05³ = 1.05 × 1.05 × 1.05 = 1.1025 × 1.05 = 1.157625, so A = 1000 × 1.157625 = 1157.625. The step-by-step approach and the formula give the same number because the formula is just the three multiplications written in one go.

For comparison, simple interest on $1,000 at 5% is 1000 × 0.05 = 50 each year, so after 3 years it is 1000 + 3 × 50 = 1150. Compounding gives 1157.625 − 1150 = 7.625, about $7.63 more. That extra is interest on interest. Year 2 earned 52.50 − 50 = 2.50 extra, and year 3 earned 55.125 − 50 = 5.125 extra. Together 2.50 + 5.125 = 7.625, which matches.

Compounding more often

Some accounts add interest monthly. The general formula is A = P(1 + r/k)^(kn), where k is the number of periods per year. With monthly compounding, k = 12: the rate per month is 0.05/12 ≈ 0.0041667, and there are 12 × 3 = 36 periods. Then A = 1000 × (1.0041667)³⁶ ≈ 1000 × 1.16147 ≈ 1161.47.

That is about $3.84 more than yearly compounding: 1161.47 − 1157.63 = 3.84. More frequent compounding helps a little, but the rate and time matter more.

How long to double

A quick estimate is the rule of 72: divide 72 by the rate written as a plain number. For a rate of 5, 72 / 5 = 14.4 years. The exact answer solves 1.05ⁿ = 2, so n = ln 2 / ln 1.05 = 0.6931 / 0.04879 ≈ 14.21 years. The rule is close, and easy to do in your head.

Try one yourself: $2,000 for 2 years

Use a rate of 0.04. Year 1: 2000 × 1.04 = 2080. Year 2: 2080 × 1.04 = 2163.20. With the formula, 1.04² = 1.0816, and 2000 × 1.0816 = 2163.20. Simple interest would give 2000 + 2 × 80 = 2160, so compounding adds 3.20, which is the second-year interest on the first year's 80: 80 × 0.04 = 3.20.

Why the interest per year keeps rising

Look at the interest column in the table: 50.00, 52.50, 55.125. Each year it is 5% of a bigger balance. The increases between them are 2.50 and then 2.625, and each of those is 0.05 times the previous interest. Because the interest itself grows by the same factor as the balance, the whole balance follows a clean pattern: multiply by 1.05, and again, and again.

How to check your answer

• Direction. The compound total should be above the simple-interest total whenever n is more than 1.

• Size. Year-on-year growth should match the rate, so each year's end value is the last one times 1.05.

• Units. Rates go into the formula as decimals: 0.05, not 5.

Common mistakes

• Using r = 5 instead of 0.05, which gives a huge wrong number.

• Multiplying 1000 × 0.05 × 3 and calling it the total. That is only the interest, and only for simple interest.

• Rounding at each step too early. Keep the extra decimals until the end.

• Mixing up (1 + r)ⁿ with 1 + rⁿ. For our numbers 1 + 0.05³ = 1.000125, nowhere near 1.157625.

• Using a yearly rate with monthly periods without dividing it by 12.

Where you will use it next

The same formula, with a negative rate, describes things that shrink by a fixed fraction each step, and the same compounding idea underlies exponential growth vs linear growth. For careful work with price rises and drops, see the guide to changes up and down. This article is maths practice, not financial advice.

Frequently asked questions

What is the difference between interest rate and growth factor?

The rate is 0.05. The factor is 1.05, the amount you multiply by each year.

Does a higher n always mean a much bigger total?

Over many years, yes. At a rate of 0.05, 30 years gives 1.05³⁰ ≈ 4.32, over four times the start. Compare with simple interest, which gives only 1 + 30 × 0.05 = 2.5 times.