To determine the value of the money five years ago based on an annual interest rate of 5%, you need to use the formula for calculating the present value (PV). This is derived from the future value (FV) formula as follows:
[tex]\[ \text{Future Value} = \text{Present Value} \times (1 + \text{interest rate})^{\text{years}} \][/tex]
Given the following:
- Future Value (FV) = \[tex]$10,000
- Interest Rate (i) = 5% or 0.05
- Time (t) = 5 years
We need to rearrange the formula to solve for the Present Value (PV):
\[ \text{Present Value} = \frac{\text{Future Value}}{(1 + \text{interest rate})^{\text{years}}} \]
Substituting the given values into the formula:
\[ \text{Present Value} = \frac{10,000}{(1 + 0.05)^5} \]
Calculate the denominator first:
\[ 1 + 0.05 = 1.05 \]
Raise this to the power of 5:
\[ 1.05^5 \approx 1.27628 \]
Now, divide the future value by this amount:
\[ \text{Present Value} = \frac{10,000}{1.27628} \approx 7835.26 \]
Therefore, the value of Pierce's money five years ago was approximately \$[/tex]7,835.26. Comparing this with the provided potential answers:
- A. \[tex]$7,462.15
- B. \$[/tex]7,835.26
- C. \[tex]$8,548.04
The correct answer is:
B. \$[/tex]7,835.26
