Lesson 116: Solving Simple and Compound Interest Problems

New Concept

Compound interest is interest that is paid on both the principal and on previously-earned interest. The formula is:

What’s next

Next, you’ll apply these formulas to calculate interest, compare investment strategies, and see how time is the most powerful variable in your financial future.

Property

Money that is borrowed or invested is called principal.

Explanation

Think of principal as your starting player in a game of making money. Whether you're borrowing it for a big purchase or investing it to grow, the principal is the original amount of cash you begin with. It's the foundation upon which all the interest, whether you're paying it or earning it, is calculated. Every financial story starts here!

Examples

• If you take out a loan for 5000 dollars to buy a car, the principal is 5000 dollars.
• If you open a savings account by depositing 300 dollars, your principal is 300 dollars.
• Investing 1000 dollars in a stock means your principal investment is 1000 dollars.

Property

Simple interest is interest paid on the principal only. To find simple interest, use the formula $I = Prt$.

Explanation

Imagine you plant a money tree that only grows fruit on its original branches. That’s simple interest! It calculates earnings only on your starting amount (the principal). It’s a steady, predictable way to grow your money, adding the same fixed amount each period. It’s simple, straightforward, and easy to calculate, but it doesn't have that explosive growth power.

Examples

• A 2000 dollars investment at $4\%$ simple interest for 5 years earns $I = 2000(0.04)(5) = 400$ dollars in interest.
• To find the total after 3 years on a 1000 dollars loan at $7\%$ simple interest: $I = 1000(0.07)(3) = 210$ dollars. The total owed is $1000 + 210 = 1210$ dollars.
• If you earned 600 dollars on a 4000 dollars investment over 3 years, the rate was $600 = 4000 \cdot r \cdot 3$, so $r = 0.05$ or $5\%$.

Simple interest is straightforward, but let's walk through finding the total amount, which is a common final step. This example will use the key idea of the simple interest formula.

Example Problem

$15,000 is invested for 12 years at$3\%$ simple interest. How much money will be in the account after 12 years?

Step-by-Step

1. First, we need to find the interest earned. We'll use the simple interest formula, $I = Prt$.
2. The principal $P$ is $15,000$. The rate $r$ is $3\%$, which we write as the decimal $0.03$. The time $t$ is $12$ years.
3. Substitute these values into the formula:

4. Simplify the expression to find the interest:

5. The account will earn $5,400$ dollars in interest. The question asks for the total amount in the account, so we add this interest to the original principal.

6. There will be $20,400$ dollars in the account after 12 years.

Property

Compound interest is interest that is paid on both the principal and on previously-earned interest. The formula is $A = P \left(1 + \frac{r}{n}\right)^{nt}$.

Explanation

This is where the real magic happens! Think of compound interest as a snowball rolling downhill. It picks up more snow, gets bigger, and then picks up even more snow because it’s bigger. You earn interest on your original principal AND on the interest you've already accumulated. This chain reaction makes your money grow exponentially faster over time!

Examples

• Investing 3000 dollars at $5\%$ compounded annually for 10 years yields $A = 3000(1 + 0.05)^{10} \approx 4886.68$ dollars.
• The same 3000 dollars at $5\%$ compounded quarterly for 10 years yields $A = 3000(1 + \frac{0.05}{4})^{4 \cdot 10} \approx 4930.86$ dollars.
• A 1000 dollars credit card purchase at $20\%$ compounded monthly for 1 year becomes $A = 1000(1 + \frac{0.20}{12})^{12 \cdot 1} \approx 1219.39$ dollars owed.

Let's see how changing the compounding frequency, a key idea of compound interest, can supercharge an investment's growth.

Example Problem

$6,000 is invested at$5\%$ compounded quarterly. Find the value of the investment after 8 years.

Step-by-Step

1. We will use the compound interest formula:
   $$A = P\left(1 + \frac{r}{n}\right)^{nt}$$
2. The principal $P$ is $6,000$. The rate $r$ is $5\%$ or $0.05$. The time $t$ is $8$ years. Since interest is compounded quarterly, $n=4$.
3. Substitute the values into the formula:

4. Now, use the order of operations to simplify. First, handle the terms inside the parentheses and the exponent.

5. Next, simplify the power. It's best to use a calculator for this and not round the intermediate result.

6. Finally, multiply to find the total amount and round to the nearest cent.

7. The value of the investment will be $8,928.78$ dollars.