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Chapter 7: Problem 18

Floyd borrowed 1000 dollar for 2 years 3 months at \(8 \%\) simple interest. a. How much interest will Floyd have to pay? b. What will be the total amount that he has to pay back?

Short Answer

Expert verified
a. 180 dollars. b. 1180 dollars.

Step by step solution

01

Understand the Simple Interest Formula

The formula for simple interest is given by: \[ I = P \times R \times T \] where, - \(I\) is the interest, - \(P\) is the principal amount, - \(R\) is the rate of interest per year, - \(T\) is the time in years.
02

Identify the Given Values

From the problem, the given values are: - Principal amount, \(P = 1000\) dollars, - Rate of interest, \(R = 8\text{\%} = 0.08\), - Time, \(T = 2 \text{ years } 3 \text{ months}\).
03

Convert Time into Years

Since the interest rate is per year, the time should be in years. 3 months is \( \frac{3}{12} = 0.25 \) years. Thus, \( T = 2 + 0.25 = 2.25 \) years.
04

Calculate the Interest

Using the simple interest formula: \[ I = P \times R \times T \] Substitute the values: \[ I = 1000 \times 0.08 \times 2.25 \]Calculate: \[ I = 1000 \times 0.18 = 180 \text{ dollars} \] Therefore, Floyd will have to pay 180 dollars in interest.
05

Calculate the Total Amount to Pay Back

The total amount Floyd has to pay back is the sum of the principal amount and the interest: \[ A = P + I \] Substitute the values: \[ A = 1000 + 180 = 1180 \text{ dollars} \] Therefore, Floyd will have to pay back a total of 1180 dollars.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

interest computation
Simple interest is a straightforward method to calculate the interest on a loan or investment. It does not take into account the effect of compounding. The formula for calculating simple interest is given by: \( I = P \times R \times T \) where:
  • \( I \) is the interest amount.
  • \( P \) is the principal amount (the initial amount of money).
  • \( R \) is the annual interest rate (expressed as a decimal).
  • \( T \) is the time the money is invested or borrowed for, in years.
To apply this formula, plug in the given values for the variables, and perform the arithmetic operations. In this case, after converting the months into years, multiply them by the principal and the rate to get the interest. This value represents the extra money Floyd needs to pay on top of the borrowed amount.
principal amount
The principal amount refers to the original sum of money borrowed or invested, before any interest. In the exercise example, Floyd borrowed \( \$1000 \). When calculating interest, the principal amount is a key component, as it determines how much interest will accumulate over time. Larger principal amounts will result in higher interest when all other factors remain constant. The principal amount is used directly in the simple interest formula. Always ensure to identify this amount correctly, especially in problems where multiple figures are mentioned.
time conversion in years
Interest rates are typically expressed annually, so time must be converted into years before using it in the simple interest formula. Many problems, like this one, involve periods that are not whole years. For example, Floyd borrowed money for 2 years and 3 months. Since there are 12 months in a year: Three months can be converted as follows: \( \frac{3}{12} = 0.25 \) years. So, adding this to the 2 years gives a total time of \( 2 + 0.25 = 2.25 \) years. Understanding this conversion is crucial for accurate interest calculation. Always break down the time component into yearly fractions to ensure precise computations.
total payment calculation
After calculating the interest, the next step is to determine the total amount Floyd has to repay. This total amount includes both the principal and the interest. Use the formula: \( A = P + I \) where:
  • \( A \) is the total amount to be paid.
  • \( P \) is the principal amount.
  • \( I \) is the interest calculated.
For Floyd, the total amount becomes: \( 1000 + 180 = 1180 \) dollars. The calculation is simple鈥攁dd the interest to the principal amount to find the total repayment amount. This shows the overall cost of borrowing, which is essential in financial planning.

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