Calculating Compound Interest

Welcome to Lesson 4 of 5 in Interest, Savings, and Borrowing — we are nearly at the finish line! So far, we have built up from reading interest terms to computing single-period interest and then extending that work across multiple periods with simple interest. In every calculation up to this point, the base stayed locked at the original principal.

This lesson changes the game. We will explore compound interest, where each period's interest is added to the balance before the next calculation begins. Instead of a fixed base, the base grows from one period to the next. We will work through this process year by year, building each new balance step by step.

With simple interest, the base never moves — it is always the original deposit or loan amount. Compound interest takes a different approach. At the end of each period, the interest earned is folded into the balance, and that updated balance becomes the base for the following period.

Think of it like a snowball rolling downhill. Each rotation picks up a bit more snow, which makes the ball bigger, which means the next rotation picks up even more. With compound interest, each period's interest is a little larger than the last because the base itself has grown. This "interest on interest" effect is what makes compounding powerful over time.

Let's trace through a concrete example. Suppose we deposit $1,000 into a savings account that pays 10% annual interest, compounded yearly. We want to find the balance at the end of each year.

Year 1 looks exactly like a single-period calculation. We multiply the principal by the rate:

The new balance at the end of Year 1 is:

So far, this matches what simple interest would give us. The difference shows up in the very next step.

Here is where compounding parts ways with simple interest. Instead of going back to the original $1,000, we use the end-of-Year-1 balance of $1,100 as the base for Year 2:

Notice that Year 2's interest is $110, not $100. The extra $10 comes from earning interest on the $100 that was added after Year 1. Under simple interest, the Year 2 balance would be $1,200. With compounding, it is $1,210. The gap is small at first, but it widens with each passing period.