Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Discover the answers you need from a community of experts ready to help you with their knowledge and experience in various fields. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
To determine which expression calculates the same present value (PV) from the given data, let's examine each option provided.
Given data:
- [tex]\( N = 96 \)[/tex] (total number of payments)
- Annual interest rate: [tex]\( 5.4\% \)[/tex] converted to a decimal [tex]\( = 0.054 \)[/tex]
- [tex]\( PMT = -560 \)[/tex]
- [tex]\( FV = 0 \)[/tex]
- [tex]\( P / Y = 12 \)[/tex] (payments per year)
- [tex]\( C / Y = 12 \)[/tex] (compounding periods per year)
- Payments occur at the end of the period: PMT:END.
1. Convert the annual interest rate to a monthly interest rate:
[tex]\[ \text{Monthly interest rate} = \frac{\text{Annual rate}}{P / Y} = \frac{0.054}{12} = 0.0045 \][/tex]
2. Confirm the number of payments:
[tex]\[ N = 8 \times 12 = 96 \][/tex] (as already given)
Next, let's evaluate each expression with the given data:
### Option A:
[tex]\[ \frac{560\left((1+0.054)^{0}-1\right)}{(0.054)(1+0.054)^{0}} \][/tex]
[tex]\[ = \frac{560 \left(1 - 1\right)}{0.054 \cdot 1} = \frac{560 \cdot 0}{0.054} = 0 \][/tex]
### Option B:
[tex]\[ \frac{560\left((1+0.0045)^{96}-1\right)}{(0.0045)(1+0.0045)^{96}} \][/tex]
[tex]\[ = \frac{560 \left((1+0.0045)^{96} - 1\right)}{0.0045 \cdot (1+0.0045)^{96}} \][/tex]
The calculation of this expression will result in a negative value which is approximately:
[tex]\[ \approx -43,575.61 \][/tex]
### Option C:
[tex]\[ \frac{560\left((1+0.054)^0 -1\right)}{(0.054)(1+0.054)^6} \][/tex]
[tex]\[ = \frac{560 \left(1 - 1\right)}{(0.054)(1+0.054)^6} = \frac{560 \cdot 0}{0.054 \cdot (1.3561)} = 0 \][/tex]
### Option D:
[tex]\[ \frac{560\left((1+0.0045)^8 -1\right)}{(0.0045)(1+0.0045)^8} \][/tex]
[tex]\[ = \frac{560 \left( (1+0.0045)^8 -1 \right)}{(0.0045)(1+0.0045)^8} \][/tex]
The calculation of this expression will result in a negative value which is approximately:
[tex]\[ \approx -4,390.62 \][/tex]
### Conclusion:
The expression that returns a value for PV that matches is Option B.
[tex]\[ \frac{560\left((1+0.0045)^{96}-1\right)}{(0.0045)(1+0.0045)^{96}} \approx -43,575.61 \][/tex]
Option B is the correct answer.
Given data:
- [tex]\( N = 96 \)[/tex] (total number of payments)
- Annual interest rate: [tex]\( 5.4\% \)[/tex] converted to a decimal [tex]\( = 0.054 \)[/tex]
- [tex]\( PMT = -560 \)[/tex]
- [tex]\( FV = 0 \)[/tex]
- [tex]\( P / Y = 12 \)[/tex] (payments per year)
- [tex]\( C / Y = 12 \)[/tex] (compounding periods per year)
- Payments occur at the end of the period: PMT:END.
1. Convert the annual interest rate to a monthly interest rate:
[tex]\[ \text{Monthly interest rate} = \frac{\text{Annual rate}}{P / Y} = \frac{0.054}{12} = 0.0045 \][/tex]
2. Confirm the number of payments:
[tex]\[ N = 8 \times 12 = 96 \][/tex] (as already given)
Next, let's evaluate each expression with the given data:
### Option A:
[tex]\[ \frac{560\left((1+0.054)^{0}-1\right)}{(0.054)(1+0.054)^{0}} \][/tex]
[tex]\[ = \frac{560 \left(1 - 1\right)}{0.054 \cdot 1} = \frac{560 \cdot 0}{0.054} = 0 \][/tex]
### Option B:
[tex]\[ \frac{560\left((1+0.0045)^{96}-1\right)}{(0.0045)(1+0.0045)^{96}} \][/tex]
[tex]\[ = \frac{560 \left((1+0.0045)^{96} - 1\right)}{0.0045 \cdot (1+0.0045)^{96}} \][/tex]
The calculation of this expression will result in a negative value which is approximately:
[tex]\[ \approx -43,575.61 \][/tex]
### Option C:
[tex]\[ \frac{560\left((1+0.054)^0 -1\right)}{(0.054)(1+0.054)^6} \][/tex]
[tex]\[ = \frac{560 \left(1 - 1\right)}{(0.054)(1+0.054)^6} = \frac{560 \cdot 0}{0.054 \cdot (1.3561)} = 0 \][/tex]
### Option D:
[tex]\[ \frac{560\left((1+0.0045)^8 -1\right)}{(0.0045)(1+0.0045)^8} \][/tex]
[tex]\[ = \frac{560 \left( (1+0.0045)^8 -1 \right)}{(0.0045)(1+0.0045)^8} \][/tex]
The calculation of this expression will result in a negative value which is approximately:
[tex]\[ \approx -4,390.62 \][/tex]
### Conclusion:
The expression that returns a value for PV that matches is Option B.
[tex]\[ \frac{560\left((1+0.0045)^{96}-1\right)}{(0.0045)(1+0.0045)^{96}} \approx -43,575.61 \][/tex]
Option B is the correct answer.
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
