Problem 59 - Precalculus Student Solutions Manual 5th

Question

Simple Interest Elmer Velasquez borrowed \(\$ 3150\) from his brother Julio to pay for books and tuition. He agreed to repay Julio in 6 months with simple annual interest at 4 \(\%\). (a) How much will the interest amount to? (b) What amount must Elmer pay Julio at the end of the 6 months?

Step-by-Step Solution

Answer

The interest amount is \( \$ 63 \). Elmer must pay \( \$ 3213 \) at the end of 6 months.

Step 1: Understand the Simple Interest Formula

The simple interest formula is given by: \[ I = P \times r \times t \] where \( I \) is the interest,\( P \) is the principal amount,\( r \) is the annual interest rate, and \( t \) is the time in years.

Step 2: Convert Time to Years

Since time needs to be in years and Elmer agreed to repay in 6 months, convert 6 months to years. \[ t = \frac{6}{12} = 0.5 \text{ years} \]

Step 3: Substitute the Values into the Formula

Now, substitute the given values into the simple interest formula. \[ P = 3150, \quad r = 0.04, \quad t = 0.5 \] \[ I = 3150 \times 0.04 \times 0.5 \]

Step 4: Calculate the Interest

Compute the interest amount. \[ I = 3150 \times 0.04 \times 0.5 = 63 \] The interest amount is \( \$ 63 \).

Step 5: Compute the Total Amount to Repay

To find the total amount Elmer must pay at the end of 6 months, add the interest amount to the principal. \[ \text{Total Amount} = P + I = 3150 + 63 = 3213 \] Elmer must pay \( \$ 3213 \) to Julio.

Key Concepts

Principal Amount

In the context of simple interest calculation, the principal amount is the initial sum of money that is borrowed or invested. It is crucial because it serves as the foundation upon which interest is calculated. In our example, Elmer borrowed a principal amount of \( \$3150\) from his brother Julio.

Understanding the principal amount is straightforward, but it's essential to grasp fully as it directly impacts how much interest you'll incur or earn. In formula terms, it's represented by the symbol \(P\). Here’s a quick recap on its importance:

• It is the baseline for interest calculation.
• An increase in the principal amount means more interest.
• It remains constant if the interest is simple, meaning it doesn’t change over time.

Maintain clarity on the principal amount before diving into more complex calculations like interest and total repayment.

Annual Interest Rate

The annual interest rate is the percentage of the principal amount that is paid as interest over a period of one year. For accurate calculations, this rate must be expressed as a decimal. In our example, Elmer agreed to an annual interest rate of 4\% or \(0.04\).

Here’s how you can break it down:

• Convert percentage to a decimal by dividing by 100. (i.e., \%4 = 0.04)
• The rate is used in the interest formula to compute how much interest is generated annually.
• It is a fixed rate in simple interest calculations, making it predictable and easier to compute.

Understanding the annual interest rate is crucial since any change in this rate significantly affects the interest accrued over time.

Time Conversion

In simple interest calculations, 'time' must always be in years. However, often, durations might be provided in different units like months or days. Therefore, it is necessary to convert these durations into years to apply them correctly in the simple interest formula.

For Elmer's repayment plan of 6 months, we converted the time to years as follows: \[t = \frac{6}{12} = 0.5 \text{ years} \]

• Always divide the number of months by 12 to convert to years.
• If it’s days, divide by 365 (or 366 for leap years).
• Precise time conversion ensures accurate interest calculations.

In our example, upon converting 6 months to 0.5 years, it allowed us to use the simple interest formula correctly. Proper time conversion is pivotal for achieving accurate financial computations.