Welcome back to Interest, Savings, and Borrowing! You are now on Lesson 3 of 5, which means you are more than halfway through this course. In the first two lessons, we learned how to read the key components of a financial statement and then how to calculate interest for a single period. Those skills gave us the building blocks; now we are ready to stack them up.

In this lesson, we will extend the one-period calculation to multiple periods. We will see how simple interest accumulates over two, three, or more years, and we will learn how to find both the total interest and the ending balance. The central idea is surprisingly straightforward: every single period uses the original principal as its base.

Before we jump into the math, let's build some intuition. Imagine you lend a friend $1,000 and agree on 10% simple interest per year. After one year, your friend owes $100 in interest. What about year two?

With simple interest, the answer is another $100 — not $110. The interest charge is always calculated on the original $1,000, regardless of how much interest has already piled up. The $100 earned in year one does not become part of the base for year two.

This "same base every time" rule is the defining feature of simple interest, and it is what makes the multi-period calculation so predictable. Each year adds the exact same dollar amount of interest because the base never grows.

The one-period interest formula from the previous lesson is $I = P \times r$. For simple interest over multiple periods, we repeat that same calculation for each period, always plugging in the original principal $P$.

Let's trace through a concrete example. Suppose we deposit $4,000 into an account that pays 5% annual simple interest and we leave it there for 3 years.

Notice how the "Base Used" column never changes. Each year produces the same $200 because the base is always the original $4,000. After 3 years, the total interest is $600, and the ending balance is $4,000 + $600 = $4,600.

Because every year's interest is identical, we do not need to list each year separately. We can multiply the one-period interest by the number of periods to get the total in one step:

Here $P$ is the original principal, $r$ is the annual rate as a decimal, and $$ is the number of time periods (years, in our examples). Applying this to the savings example above:

Let's apply the formula to a real-world loan. Suppose you borrow $3,000 to cover an emergency car repair, and the lender charges 6% annual simple interest over a 4-year repayment term. We want to know the total interest owed and the full amount you must repay.

Step 1 — Convert the rate: $6\% = 0.06$.

Step 2 — Calculate the total interest:

The most common errors in multi-period problems are specific to the way $t$ interacts with the rest of the formula. Keep these pointers in mind as you practice:

• Always use the original principal as the base. It is tempting to add each year's interest to the balance before computing the next year. That approach would give you compound interest, which we will explore in the next lesson. For simple interest, the base stays fixed.
• Make sure $t$ matches the rate's time unit. Because $r$ is an annual rate in our problems, $t$ must be in years. A 2-year period means $t = 2$, not (months).

In this lesson, we learned that simple interest over multiple periods is calculated with the formula $I = P \times r \times t$. The key insight is that every period's interest is based on the original principal, so the interest earned each period stays the same. The ending balance is then simply $P + I$.

Up next, you will put these ideas to work through a series of hands-on exercises. You will fill in year-by-year tables, compute multi-period totals for different principal-and-rate pairs, and tackle a realistic loan scenario entirely on your own.