Questions: Find the periodic payment R required to accumulate a sum of S dollars over t years with interest earned at the rate of r % / year compounded m times a year. (Round your answer to the nearest cent.) S=360,000, r=2.2, t=12, m=6

Solution Steps

To find the periodic payment \( R \) required to accumulate a sum of \( S \) dollars over \( t \) years with interest earned at the rate of \( r \% \) per year compounded \( m \) times a year, we can use the future value of an ordinary annuity formula:

\[ S = R \left( \frac{(1 + \frac{r}{100m})^{mt} - 1}{\frac{r}{100m}} \right) \]

Rearranging to solve for \( R \):

\[ R = \frac{S \cdot \frac{r}{100m}}{(1 + \frac{r}{100m})^{mt} - 1} \]

Given:

• \( S = 360,000 \)
• \( r = 2.2 \)
• \( t = 12 \)
• \( m = 6 \)

We can plug these values into the formula to find \( R \).

We are given the following values:

• \( S = 360,000 \)
• \( r = 2.2 \)
• \( t = 12 \)
• \( m = 6 \)

Convert the annual interest rate from a percentage to a decimal: \[ r_{\text{decimal}} = \frac{r}{100} = \frac{2.2}{100} = 0.022 \]

Using the formula for the periodic payment \( R \): \[ R = \frac{S \cdot \frac{r_{\text{decimal}}}{m}}{(1 + \frac{r_{\text{decimal}}}{m})^{mt} - 1} \] Substituting the values: \[ R = \frac{360,000 \cdot \frac{0.022}{6}}{(1 + \frac{0.022}{6})^{6 \cdot 12} - 1} \]

Calculating the components:

• \( \frac{0.022}{6} = 0.0036667 \)
• \( 1 + 0.0036667 = 1.0036667 \)
• \( mt = 6 \cdot 12 = 72 \)
• \( (1.0036667)^{72} \approx 1.2996 \)

Now substituting back: \[ R = \frac{360,000 \cdot 0.0036667}{1.2996 - 1} = \frac{1,320.012}{0.2996} \approx 4,378.11 \]

Final Answer