Compound Interest

Compound interest grows a starting amount by paying interest on the whole balance each year, including interest already earned.

A = P(1 + r)^t

Each year grows a little faster than the last, because interest earns interest.

What each symbol means

Compound interest is the reason saving early is so powerful. Unlike simple interest, which always uses the starting amount, compound interest pays interest on the whole balance, including the interest you already earned. So the growth speeds up over time.

The formula (for interest added once a year) is A = P(1 + r)^t. It gives you the final total directly, not just the interest.

This gives the final total: The simple-interest formula gives just the interest. This one gives the whole final amount, A. To find only the interest earned, subtract the principal: interest = A - P.

A worked example

\$1000 at 10% compounded yearly for 2 years

  1. Write the formula: A = P(1 + r)^t.
  2. Change the rate to a decimal: 10\% = 0.10.
  3. Put in the numbers: A = 1000 × (1 + 0.10)².
  4. Add inside the brackets: 1 + 0.10 = 1.10.
  5. Apply the power: 1.10² = 1.21.
  6. Multiply: A = 1000 × 1.21 = 1210.
  7. You end with \1210. The interest earned is1210 - 1000 = 210$ dollars.

Why it beats simple interest: Simple interest on the same deposit would earn \100 per year, so \200 in 2 years. Compound earned \210. The extra \10 is interest paid on the first year's interest. Over many years, that gap grows a lot.

Why the power appears

  1. After one year, the balance is the principal plus its interest: P + Pr = P(1 + r).
  2. The second year earns interest on that new, larger balance, so multiply by (1 + r) again: P(1 + r)(1 + r) = P(1 + r)².
  3. Each year multiplies by another (1 + r).
  4. After t years you have multiplied by (1 + r) a total of t times, which is P(1 + r)^t. That repeated multiplication is exactly what the power means.

Practice questions

Question 1

Find the final amount for \$1000 at 10% compounded yearly for 1 year.

  1. Write the formula: A = P(1 + r)^t.
  2. Change the rate to a decimal: 10\% = 0.10, so 1 + r = 1.10.
  3. Put in the numbers: A = 1000 × 1.10^1.
  4. A power of 1 changes nothing, so A = 1000 × 1.10 = 1100.
  5. The final amount is \$1100.

Answer: \$1100

Question 2

Find the final amount for \$2000 at 5% compounded yearly for 2 years.

  1. Write the formula: A = P(1 + r)^t.
  2. Change the rate to a decimal: 5\% = 0.05, so 1 + r = 1.05.
  3. Apply the power: 1.05² = 1.1025.
  4. Multiply by the principal: A = 2000 × 1.1025 = 2205.
  5. The final amount is \$2205.

Answer: \$2205

Question 3

How much interest is earned on \$500 at 10% compounded yearly for 2 years?

  1. First find the final amount with A = P(1 + r)^t.
  2. Here 1 + r = 1.10, so 1.10² = 1.21.
  3. Multiply: A = 500 × 1.21 = 605.
  4. The interest is the final amount minus the principal: 605 - 500 = 105.
  5. So the interest earned is \$105.

Answer: \$105

Question 4

\$1000 grows at 20% compounded yearly. What is the balance after 2 years?

  1. Write the formula: A = P(1 + r)^t.
  2. Change the rate to a decimal: 20\% = 0.20, so 1 + r = 1.20.
  3. Apply the power: 1.20² = 1.44.
  4. Multiply by the principal: A = 1000 × 1.44 = 1440.
  5. The balance after 2 years is \$1440.

Answer: \$1440

Question 5

Over 1 year, do simple and compound interest give the same result?

  1. Compound interest only pulls ahead once earned interest starts earning its own interest.
  2. That first happens in the second year, when the new balance is larger.
  3. In the very first year there is no earlier interest to build on yet.
  4. So over a single year the two methods give exactly the same result.

Answer: Yes