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Problem 5-4: Calculating Interest Rates [LO3]

Solve for the unknown interest rate in each of the following. (Do not round intermediate calculations and enter your answers as a percent rounded to 2 decimal places, e.g., 32.16.)

\begin{tabular}{|r|r|r|r|r|}
\hline
\textbf{Present Value} & \textbf{Years} & \multicolumn{2}{|c|}{\textbf{Interest Rate}} & \textbf{Future Value} \\
\hline
\[tex]$220 & 4 & 5.29 & \% & \$[/tex]270 \\
\hline
\[tex]$340 & 18 & 6.05 & \% & \$[/tex]986 \\
\hline
\[tex]$37,000 & 19 & & \% & \$[/tex]169,819 \\
\hline
\[tex]$36,261 & 25 & & \% & \$[/tex]169,819 \\
\hline
\end{tabular}



Answer :

GinnyAnswer
Certainly! Let's solve for the unknown interest rates for the given scenarios using the future value (FV) and present value (PV) formulas.

### Formula:
The formula to calculate the future value (FV) from the present value (PV) with a certain interest rate [tex]\( r \)[/tex] over [tex]\( n \)[/tex] years is:
[tex]\[ FV = PV \times (1 + r)^n \][/tex]

To find the interest rate, [tex]\( r \)[/tex], we rearrange the formula:
[tex]\[ r = \left(\frac{FV}{PV}\right)^{\frac{1}{n}} - 1 \][/tex]

We'll solve each part as follows:

### 1. [tex]\( \$ 220 \)[/tex], 4 years, future value 5.29%
For the first case, we are given the interest rate of 5.29%.

### 2. [tex]\( \$ 340 \)[/tex], 18 years, future value [tex]\( \$ 270 \)[/tex]
Given:
- [tex]\( PV = \$ 340 \)[/tex]
- [tex]\( n = 18 \)[/tex]
- [tex]\( FV = \$ 270 \)[/tex]

Using the formula:
[tex]\[ r = \left(\frac{270}{340}\right)^{\frac{1}{18}} - 1 \][/tex]

The interest rate [tex]\( r \)[/tex] can be calculated as approximately:
[tex]\[ r \approx -1.27\% \][/tex]

### 3. [tex]\( \$ 37,000 \)[/tex], 19 years, future value [tex]\( \$ 986 \)[/tex]
Given:
- [tex]\( PV = \$ 37,000 \)[/tex]
- [tex]\( n = 19 \)[/tex]
- [tex]\( FV = \$ 986 \)[/tex]

Using the formula:
[tex]\[ r = \left(\frac{986}{37,000}\right)^{\frac{1}{19}} - 1 \][/tex]

The interest rate [tex]\( r \)[/tex] can be calculated as approximately:
[tex]\[ r \approx -17.37\% \][/tex]

### 4. [tex]\( \$ 36,261 \)[/tex], 25 years, future value [tex]\( \$ 169,819 \)[/tex]
Given:
- [tex]\( PV = \$ 36,261 \)[/tex]
- [tex]\( n = 25 \)[/tex]
- [tex]\( FV = \$ 169,819 \)[/tex]

Using the formula:
[tex]\[ r = \left(\frac{169,819}{36,261}\right)^{\frac{1}{25}} - 1 \][/tex]

The interest rate [tex]\( r \)[/tex] can be calculated as approximately:
[tex]\[ r \approx 6.37\% \][/tex]

### Summary:
- For [tex]\( \$ 220 \)[/tex], 4 years, the interest rate is 5.29%.
- For [tex]\( \$ 340 \)[/tex], 18 years, the interest rate is [tex]\( -1.27\% \)[/tex].
- For [tex]\( \$ 37,000 \)[/tex], 19 years, the interest rate is [tex]\( -17.37\% \)[/tex].
- For [tex]\( \$ 36,261 \)[/tex], 25 years, the interest rate is [tex]\( 6.37\% \)[/tex].

Note that negative interest rates indicate a decrease in value over time.