6.4 Solve Simple Interest Applications

Simple Interest Applications

Simple interest is a way to calculate how much money you earn (or owe) on an investment or loan over time. It depends on three things: how much money you start with, the interest rate, and how long the money sits. This formula shows up constantly in real-world finance, so it's worth getting comfortable with it now.

Simple Interest Formula

The core formula is:

$I = Prt$

Here's what each variable means:

• $I$ = interest earned (or owed)
• $P$ = principal, the initial amount invested or borrowed
• $r$ = annual interest rate written as a decimal (so 5% becomes 0.05)
• $t$ = time in years

Finding interest: Multiply all three values together. If you invest $1,000 at 5\\% for 3 years: $$I = 1{,}000 \times 0.05 \times 3 = 150$$. You'd earn$150 in interest.

The formula can also be rearranged to solve for any variable you're missing:

• Find the principal: $P = \frac{I}{rt}$
  • If you earned $150 at 5% over 3 years: $P = \frac{150}{0.05 \times 3} = 1{,}000$
• Find the rate: $r = \frac{I}{Pt}$
  • If $1,000 earned $150 over 3 years: $r = \frac{150}{1{,}000 \times 3} = 0.05$, which is 5%
• Find the time: $t = \frac{I}{Pr}$
  • If $1,000 earned $150 at 5%: $t = \frac{150}{1{,}000 \times 0.05} = 3$ years

The trick is always the same: isolate the variable you need by dividing the interest by the product of the other two.

Real-World Financial Applications

Simple interest shows up in savings accounts, certificates of deposit (CDs), and many short-term loans.

• For a savings account, the principal is your initial deposit, the rate is the annual percentage yield (APY), and the time is how long you leave the money in.
• For a loan, the principal is the amount you borrow, the rate is the annual percentage rate (APR), and the time is how long you take to repay it.

Example: You borrow $5,000 at 6\\% APR for 2 years. The interest you owe is $$I = 5{,}000 \times 0.06 \times 2 = 600$$. So you'd pay back the original$5,000 plus $600 in interest, for a total of $5,600.

Time Unit Conversions

The rate in the formula is annual, so the time must also be in years. If you're given months or days, you need to convert.

• Months to years: Divide by 12. For example, 6 months = $\frac{6}{12} = 0.5$ years.
• Days to years: Divide by 365. For example, 90 days = $\frac{90}{365} \approx 0.247$ years. (Some banks use 360 days instead of 365, so check the problem.)

Example: You deposit $2,000 at 4\\% simple interest for 9 months. First convert: $$t = \frac{9}{12} = 0.75$$ years. Then calculate: $$I = 2{,}000 \times 0.04 \times 0.75 = 60$$. You'd earn$60 in interest.

Beyond Simple Interest (Preview)

As you move forward in math and finance, you'll encounter more complex ideas built on top of simple interest:

• Compound interest calculates interest on both the original principal and any interest already earned. This makes money grow faster over time.
• Future value is the total amount an investment will be worth at a later date.
• Present value is what a future sum of money is worth right now.
• Amortization is how loans get broken into equal payments over time (like a car payment or mortgage).

You don't need to calculate these yet, but recognizing the terms is helpful. For now, focus on mastering the simple interest formula and its rearrangements.