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The following group of values was entered into the TVM Solver of a graphing calculator:

N = 96
I% = 5.4
PV = ?
PMT = -560
FV = 0
P/Y = 12
C/Y = 12
PMT: END

Which of these expressions will return the same value for PV?

A. [tex]\(\frac{(560)\left((1+0.054)^{00}-1\right)}{(0.054)(1+0.054)^{00}}\)[/tex]

B. [tex]\(\frac{(\$ 560)\left((1+0.0045)^{n0}-1\right)}{(0.0045)(1+0.0045)^{10}}\)[/tex]

C. [tex]\(\frac{(560)\left((1+0.054)^0-1\right)}{(0.054)(1+0.054)^6}\)[/tex]

D. [tex]\(\frac{(\$ 560)\left((1+0.0045)^8-1\right)}{(0.0045)(1+0.0045)^8}\)[/tex]

Sagot :

GinnyAnswer
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.
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