Find the monthly payment R needed to have a sinking fund accumulate the future value A . The yearly interest rate r and the time t in years are given below. Interest is compounded
monthly.

(Round up to the nearest cent.)

Ask by Moran Hammond. in the United States

Mar 05,2025

Question

Find the monthly payment R needed to have a sinking fund accumulate the future value A . The yearly interest rate r and the time t in years are given below. Interest is compounded
monthly.

(Round up to the nearest cent.)

Ask by Moran Hammond. in the United States

Mar 05,2025

UpStudy AI · Accepted Answer

To find the monthly payment $$ R $$ needed to accumulate a future value $$ A $$ in a sinking fund, we can use the formula for the future value of a series of payments (ordinary annuity) compounded at a certain interest rate. The formula is given by:

$$
A = R \times \frac{(1 + i)^{nt} - 1}{i}
$$

Where:
- $$ A $$ is the future value of the sinking fund.
- $$ R $$ is the monthly payment.
- $$ i $$ is the monthly interest rate (annual rate divided by 12).
- $$ n $$ is the number of compounding periods per year (12 for monthly).
- $$ t $$ is the number of years.

### Step 1: Extract the known values
- $$ A = 6000 $$
- $$ r = 8\% = 0.08 $$
- $$ t = 2 $$

### Step 2: Calculate the monthly interest rate $$ i $$
$$
i = \frac{r}{12} = \frac{0.08}{12}
$$

### Step 3: Calculate the total number of payments $$ nt $$
$$
n = 12 \quad \text{(monthly payments)}
$$
$$
nt = 12 \times 2 = 24
$$

### Step 4: Substitute the values into the formula and solve for $$ R $$
Rearranging the formula to solve for $$ R $$:
$$
R = \frac{A \cdot i}{(1 + i)^{nt} - 1}
$$

Now, we will calculate $$ i $$ and then substitute all values into the formula to find $$ R $$.

Let's perform the calculations.

1. Calculate $$ i $$:
$$
i = \frac{0.08}{12}
$$

2. Substitute into the formula for $$ R $$:
$$
R = \frac{6000 \cdot i}{(1 + i)^{24} - 1}
$$

Now, I will calculate these values.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{0.08}{12}$$
- step1: Convert the expressions:
  $$\frac{\frac{2}{25}}{12}$$
- step2: Multiply by the reciprocal:
  $$\frac{2}{25}\times \frac{1}{12}$$
- step3: Reduce the numbers:
  $$\frac{1}{25}\times \frac{1}{6}$$
- step4: Multiply the fractions:
  $$\frac{1}{25\times 6}$$
- step5: Multiply:
  $$\frac{1}{150}$$
Calculate or simplify the expression $$ 6000*(0.08/12)/((1+(0.08/12))^24-1) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{6000\left(\frac{0.08}{12}\right)}{\left(\left(1+\left(\frac{0.08}{12}\right)\right)^{24}-1\right)}$$
- step1: Remove the parentheses:
  $$\frac{6000\left(\frac{0.08}{12}\right)}{\left(1+\left(\frac{0.08}{12}\right)\right)^{24}-1}$$
- step2: Divide the terms:
  $$\frac{6000\left(\frac{0.08}{12}\right)}{\left(1+\frac{1}{150}\right)^{24}-1}$$
- step3: Add the numbers:
  $$\frac{6000\left(\frac{0.08}{12}\right)}{\left(\frac{151}{150}\right)^{24}-1}$$
- step4: Divide the terms:
  $$\frac{6000\times \frac{1}{150}}{\left(\frac{151}{150}\right)^{24}-1}$$
- step5: Multiply the numbers:
  $$\frac{40}{\left(\frac{151}{150}\right)^{24}-1}$$
- step6: Subtract the numbers:
  $$\frac{40}{\frac{151^{24}-150^{24}}{150^{24}}}$$
- step7: Multiply by the reciprocal:
  $$40\times \frac{150^{24}}{151^{24}-150^{24}}$$
- step8: Multiply:
  $$\frac{40\times 150^{24}}{151^{24}-150^{24}}$$
### Step 1: Calculate the monthly interest rate $$ i $$
$$
i = \frac{0.08}{12} \approx 0.00666667
$$

### Step 2: Calculate the monthly payment $$ R $$
Using the rearranged formula:
$$
R = \frac{6000 \cdot 0.00666667}{(1 + 0.00666667)^{24} - 1}
$$

Calculating the denominator:
$$
(1 + 0.00666667)^{24} \approx 1.1616
$$
Thus,
$$
(1 + 0.00666667)^{24} - 1 \approx 1.1616 - 1 \approx 0.1616
$$

Now substituting back into the formula for $$ R $$:
$$
R \approx \frac{6000 \cdot 0.00666667}{0.1616} \approx \frac{40}{0.1616} \approx 247.00
$$

### Final Calculation
Now, rounding up to the nearest cent, the monthly payment $$ R $$ is approximately:
$$
R \approx 247.00
$$

Thus, the monthly payment $$ R $$ needed to accumulate the future value $$ A $$ is:
$$
\boxed{247.00}
$$
The monthly payment $$ R $$ needed is \$247.00.

Find the monthly payment R needed to have a sinking fund accumulate the future value A . The yearly interest rate r and the time t in years are given below. Interest is compounded
monthly.

(Round up to the nearest cent.)

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Find the monthly payment R needed to have a sinking fund accumulate the future value A . The yearly interest rate r and the time t in years are given below. Interest is compounded monthly. (Round up to the nearest cent.)

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Find the monthly payment R needed to have a sinking fund accumulate the future value A . The yearly interest rate r and the time t in years are given below. Interest is compounded
monthly.

(Round up to the nearest cent.)