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 FV dollars at a simple interest rate r for $$ t $$ years is given by which formula? A. $$ \mathrm{PV}=\mathrm{rt}(1+\mathrm{FV}) $$ B. PV $$ =\mathrm{FV}(1+\mathrm{rt}) $$ C. PV $$ =\frac{\mathrm{FV}}{1+r t} $$ D. PV $$ =\frac{1+r t}{F V} $$ The firm should deposit $$ \$ \square $$. (Round to the nearest cent as needed.)

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UpStudy AI · Accepted Answer

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) using simple interest.
3. **Calculate the present value (PV)** that the firm needs to deposit today.

### Step 1: Calculate the Total Future Value (FV)

The firm has ordered 55 computers at a cost of $$ \$1649 $$ each. Therefore, the total cost (FV) is calculated as follows:

$$
\text{FV} = \text{Number of Computers} \times \text{Cost per Computer} = 55 \times 1649
$$

### 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) years is given by the formula:

$$
\text{PV} = \frac{\text{FV}}{1 + rt}
$$

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.
Calculate the value by following steps:
- step0: Calculate:
  $$55\times 1649$$
- step1: Multiply the numbers:
  $$90695$$
Calculate or simplify the expression $$ 1 + 0.0274 * (8/12) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$1+0.0274\left(\frac{8}{12}\right)$$
- step1: Reduce the fraction:
  $$1+0.0274\times \frac{2}{3}$$
- step2: Multiply the numbers:
  $$1+\frac{137}{7500}$$
- step3: Reduce fractions to a common denominator:
  $$\frac{7500}{7500}+\frac{137}{7500}$$
- step4: Transform the expression:
  $$\frac{7500+137}{7500}$$
- step5: Add the numbers:
  $$\frac{7637}{7500}$$
Calculate or simplify the expression $$ 90695 / (7637/7500) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{90695}{\frac{7637}{7500}}$$
- step1: Multiply by the reciprocal:
  $$90695\times \frac{7500}{7637}$$
- step2: Multiply:
  $$\frac{90695\times 7500}{7637}$$
- step3: Multiply:
  $$\frac{680212500}{7637}$$
### Step 1: Calculate the Total Future Value (FV)

The total cost of the computers is:

$$
FV = 55 \times 1649 = 90695
$$

### Step 2: Calculate the Interest Factor

We calculated the interest factor as follows:

$$
1 + rt = 1 + 0.0274 \times \frac{8}{12} \approx 1.0182
$$

### Step 3: Calculate the Present Value (PV)

Now we can calculate the present value (PV):

$$
PV = \frac{FV}{1 + rt} = \frac{90695}{1.0182} \approx 89068.02
$$

### Final Answer

The firm should deposit approximately $$ \$89068.02 $$ today to have enough money to pay for the machines in 8 months.

Thus, the final answer is:

The firm should deposit $$ \$89068.02 $$.
The firm should deposit \$89,068.02 today to pay for the computers in 8 months.

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