To determine the break-even time for this equipment purchase, we will calculate the present value (PV) of the cash inflows for each year and compare the cumulative PV to the initial equipment cost of \[tex]$9,000.
Given:
- Equipment cost: \$[/tex]9,000
- Annual cash inflow: \[tex]$2,800
- Interest rate: 10%
- Present value factors for each year are provided
We will use the formula for present value:
\[ PV = \frac{\text{Cash inflow}}{(1 + \text{interest rate})^n} \]
where \(n\) is the year.
However, for simplicity, we will directly use the present value factors provided for each year to calculate the present value of the cash inflows.
1. Year 0:
\[ PV_0 = \$[/tex]2,800 \times 1.0000 = \[tex]$2,800 \]
2. Year 1:
\[ PV_1 = \$[/tex]2,800 \times 0.9091 = \[tex]$2,545.48 \]
3. Year 2:
\[ PV_2 = \$[/tex]2,800 \times 0.8264 = \[tex]$2,314.00 \]
4. Year 3:
\[ PV_3 = \$[/tex]2,800 \times 0.7513 = \[tex]$2,103.64 \]
5. Year 4:
\[ PV_4 = \$[/tex]2,800 \times 0.6830 = \[tex]$1,912.40 \]
6. Year 5:
\[ PV_5 = \$[/tex]2,800 \times 0.6209 = \[tex]$1,738.52 \]
To find the year in which the total present value of the cash inflows equals or exceeds the equipment cost (\$[/tex]9,000), we calculate the cumulative present value for each year:
- Cumulative PV for Year 1:
[tex]\[ \text{Total PV} = \$2,800 + \$2,545.48 = \$5,345.48 \][/tex]
- Cumulative PV for Year 2:
[tex]\[ \text{Total PV} = \$5,345.48 + \$2,314.00 = \$7,659.48 \][/tex]
- Cumulative PV for Year 3:
[tex]\[ \text{Total PV} = \$7,659.48 + \$2,103.64 = \$9,763.12 \][/tex]
At this point, we compare the cumulative PV to the equipment cost:
[tex]\[ \text{Total PV} = \$9,763.12 \][/tex]
This is more than \[tex]$9,000. Therefore, the break-even occurs between the beginning of Year 4 and the end of Year 3. But since cumulative calculation for Year 3 is greater than \$[/tex]9,000, the exact break-even time would be between the accrual point of Year 3 and Year 4.
Hence:
[tex]\[ \text{Break-even time is between four and five years.} \][/tex]
Therefore, the correct answer is:
[tex]\[ \text{Break-even time is between four and five years.} \][/tex]
