Compound Interest

Understanding the concept

Compound interest is interest calculated on the starting amount plus interest already added. As a balance grows, the same interest rate can produce a larger dollar amount each period.

This lesson uses simple, fixed examples to explain the method. Identify the quantities, choose the correct formula, and check that the units in your answer make sense.

Compound-interest formula

For a fixed annual rate compounded once per year: A = P × (1 + r)^t, where A is the ending amount, P is the principal, r is the annual rate as a decimal and t is the number of years.

Write down the original values before substituting numbers. Keep full precision during the calculation and round only when presenting the final result.

Worked example

Question: Invest $1,000 at 5% per year, compounded annually, for three years.

1. Year 1: $1,000 × 1.05 = $1,050.00.
2. Year 2: $1,050 × 1.05 = $1,102.50.
3. Year 3: $1,102.50 × 1.05 = $1,157.625, or $1,157.63 rounded.

Check the answer against the original question. An answer should include the appropriate unit or percentage where relevant.

Key points to remember

• With simple interest at 5%, the same $1,000 earns $50 each year: $1,150 after three years.
• With annual compounding, the three-year ending balance is approximately $1,157.63.
• Compounding more often changes the result; fees, taxes and variable rates can also affect real outcomes.

Common mistakes

Assuming every account compounds annually or treating an illustrative rate as a guaranteed return.

To avoid this, label each number, identify the starting amount and use a quick estimate to see whether the final answer is reasonable.

Try it yourself

What does $100 become after two years at 10% compounded annually?

Year-by-year growth table

Starting with $1,000 at a fixed 5% annual interest rate, with interest added once a year:

Values shown to the nearest cent. The calculation uses full precision until the final display; individual displayed rows may differ by a cent from calculations that round interest annually.

Simple versus compound interest

With $1,000 invested at 5% for five years, simple interest earns $50 each year, producing $1,250. Compound interest earns interest on earlier interest, producing approximately $1,276.28 when compounded annually.

How compounding frequency changes the result

Some products add interest monthly or daily rather than annually. When the stated annual rate is a nominal rate, the general formula is:

P is the initial amount, r the nominal annual rate as a decimal, n the number of compounding periods per year, and t the number of years. This assumes a constant rate and no additional deposits, withdrawals, fees or taxes.

For $1,000 at a nominal 5% rate over three years, annual compounding gives about $1,157.63, while monthly compounding gives about $1,161.47.

Visual comparison

The bars compare simple and compound balances at 5% over five years, starting from $1,000. They show the interest earned above the starting amount, not the entire account balance.

Growth factors and compounding

Use this table as a quick reference. Check which quantities and units each method requires before calculating.

More worked examples

Work through each calculation and notice how the method changes with the question.

Example 1: Grow $500 at 10% for 2 years

Method: 500 × 1.1² = 605.

Example 2: Grow $2,000 at 3% for 2 years

Method: 2,000 × 1.03² = 2,121.80.

Example 3: Compare $1,000 at 5% for 2 years

Method: Compound: 1,000 × 1.05² = 1,102.50; simple: 1,000 + 100 = 1,100.

Using this in everyday life

These examples assume a constant annual rate, annual compounding and no additional transactions. Actual account returns can be reduced by fees, tax or changing rates.

Before you begin: Identify what the question asks, write down the given information, use matching units, then estimate the result to catch mistakes.

Extra practice questions

Try these independently before expanding the answer. Show your working, not just the final number.