Lesson 3-5: Solve Simple Interest Problems

Property

To translate a simple interest word problem for a single year, identify the key components of the simple interest formula, $I = p \cdot r$:

• $I$ (Interest): The extra money earned or paid.
• $p$ (Principal): The initial amount of money invested or borrowed.
• $r$ (Interest Rate): The percent of the principal earned or paid, written as a decimal.

Examples

Property

The final account balance or total amount paid is the sum of the initial principal and the accumulated interest.

Property

To find the simple interest rate ($r$) when given the principal ($P$) and the final balance ($A$) for one period, first calculate the total interest ($I$), and then divide by the principal:

Examples

• A savings account starts with a principal of 500 dollars and has a final balance of 525 dollars after one year. The interest is $525 - 500 = 25$ dollars. The interest rate is $\frac{25}{500} = 0.05$, or $5\%$.
• A loan has a principal of 1200 dollars and a total repayment (final balance) of 1272 dollars. The interest paid is $1272 - 1200 = 72$ dollars. The interest rate is $\frac{72}{1200} = 0.06$, or $6\%$.

Explanation

When you know both the starting amount and the final balance, you can figure out the interest rate in two simple steps. First, subtract the principal from the final balance to determine the actual amount of interest earned or paid. Next, divide that interest amount by the principal to find the rate as a decimal. Finally, multiply by $100$ to express that decimal as a percentage.

Property

To find the principal ($P$) when given the final balance ($A$) and the simple interest rate ($r$) for one period, use the formula:

where $r$ is the interest rate expressed as a decimal.

Examples

• If an account has a final balance of 105 dollars after earning 5% interest, the principal is $P = \frac{105}{1 + 0.05} = \frac{105}{1.05} = 100$ dollars.
• If a loan requires a total repayment of 270 dollars with an 8% interest rate, the original loan amount (principal) is $P = \frac{270}{1 + 0.08} = \frac{270}{1.08} = 250$ dollars.
• If an investment grows to 560 dollars at a 12% interest rate, the starting principal is $P = \frac{560}{1 + 0.12} = \frac{560}{1.12} = 500$ dollars.

Explanation

The final balance represents the original principal (100%) plus the interest earned. To find the principal, first add the interest rate to 100% and convert it to a decimal. Then, divide the final balance by this combined decimal value. This reverses the process of calculating the final balance to reveal the starting amount.