An accounting firm has ordered 55 new computers at a cost of \( \$ 1649 \) each. The machines will not be delivered for 8 months. What amount could the firm deposit today in an account paying \( 2.74 \% \) simple interest to have enough money to pay for the machines in 8 months? The present value PV of a future amount of \( F V \) dollars at a simple interest rate \( r \) for \( t \) years is given by which formula? A. PV \( =\mathrm{r}(1+\mathrm{FV}) \) B. PV \( =\mathrm{FV}(1+\mathrm{rt}) \) C. PV \( =\frac{F V}{1+r t} \) D. PV \( =\frac{1+r t}{\mathrm{FV}} \) The firm should deposit \( \$ \square \). (Round to the nearest cent as needed.)

To solve this problem, we need to follow these steps: 1. **Calculate the total future value (FV)** of the computers. 2. **Identify the correct formula** for present value (PV) based on the options provided. 3. **Calculate the present value (PV)** that the firm needs to deposit today. ### Step 1: Calculate the Total Future Value (FV) The firm is ordering 55 computers at a cost of \( \$1649 \) each. Therefore, the total future value (FV) is calculated as follows: \[ FV = \text{Number of Computers} \times \text{Cost per Computer} = 55 \times 1649 \] ### Step 2: Identify the Correct Formula for Present Value (PV) The present value (PV) of a future amount (FV) at a simple interest rate (r) for time (t) is given by the formula: \[ PV = \frac{FV}{1 + rt} \] This corresponds to option C. ### Step 3: Calculate the Present Value (PV) Now we need to calculate the present value using the formula identified. We have: - \( FV = 55 \times 1649 \) - \( r = 2.74\% = 0.0274 \) - \( t = 8 \text{ months} = \frac{8}{12} \text{ years} = \frac{2}{3} \text{ years} \) Now, let's calculate \( FV \) and then \( PV \). 1. Calculate \( FV \): \[ FV = 55 \times 1649 \] 2. Calculate \( PV \): \[ PV = \frac{FV}{1 + rt} \] Now, let's perform these calculations. First, calculate \( FV \): \[ FV = 55 \times 1649 \] Next, calculate \( rt \): \[ rt = 0.0274 \times \frac{2}{3} \] Finally, calculate \( PV \): \[ PV = \frac{FV}{1 + rt} \] Let's perform these calculations. Calculate the value by following steps: - step0: Calculate: \(55\times 1649\) - step1: Multiply the numbers: \(90695\) Calculate or simplify the expression \( 0.0274 * (2/3) \). Calculate the value by following steps: - step0: Calculate: \(0.0274\times \frac{2}{3}\) - step1: Convert the expressions: \(\frac{137}{5000}\times \frac{2}{3}\) - step2: Multiply the fractions: \(\frac{137\times 2}{5000\times 3}\) - step3: Multiply the terms: \(\frac{274}{5000\times 3}\) - step4: Multiply the terms: \(\frac{274}{15000}\) - step5: Reduce the fraction: \(\frac{137}{7500}\) Calculate or simplify the expression \( 1 + 0.0182 \). Calculate the value by following steps: - step0: Calculate: \(1+0.0182\) - step1: Add the numbers: \(1.0182\) Calculate or simplify the expression \( 90695 / 1.0182 \). Calculate the value by following steps: - step0: Calculate: \(\frac{90695}{1.0182}\) - step1: Convert the expressions: \(\frac{90695}{\frac{5091}{5000}}\) - step2: Multiply by the reciprocal: \(90695\times \frac{5000}{5091}\) - step3: Multiply: \(\frac{90695\times 5000}{5091}\) - step4: Multiply: \(\frac{453475000}{5091}\) Now that we have completed the calculations, let's summarize the results: 1. **Total Future Value (FV)**: \[ FV = 55 \times 1649 = 90695 \] 2. **Interest Rate Calculation (rt)**: \[ rt = 0.0274 \times \frac{2}{3} \approx 0.0182 \] 3. **Calculating \(1 + rt\)**: \[ 1 + rt \approx 1 + 0.0182 \approx 1.0182 \] 4. **Calculating Present Value (PV)**: \[ PV = \frac{FV}{1 + rt} = \frac{90695}{1.0182} \approx 89073.86 \] Thus, the firm should deposit approximately \( \$89073.86 \). ### Final Answer The firm should deposit \( \$89073.86 \).