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Which of these groups of values plugged into the TVM Solver of a graphing calculator will return the same value for PV as the expression
[tex]\[
\frac{(\$ 415)\left((1+0.003)^{24}-1\right)}{(0.003)(1+0.003)^{24}} ?
\][/tex]

A. [tex]\(N=2 ; \text{I\%}=3.6 ; \text{PV=;} \text{PMT}=-415 ; \text{FV}=0 ; \text{P/Y}=12 ; \text{C/Y}=12 ; \text{PMT: END}\)[/tex]

B. [tex]\(N=2 ; \text{I\%}=0.3 ; \text{PV=;} \text{PMT}=-415 ; \text{FV}=0 ; \text{P/Y}=12 ; \text{C/Y}=12 ; \text{PMT: END}\)[/tex]

C. [tex]\(N=24 ; \text{I\%}=0.3 ; \text{PV=;} \text{PMT}=-415 ; \text{FV}=0 ; \text{P/Y}=12 ; \text{C/Y}=12 ; \text{PMT: END}\)[/tex]

D. [tex]\(N=24 ; \text{I\%}=3.6 ; \text{PV=;} \text{PMT}=-415 ; \text{FV}=0 ; \text{P/Y}=12 ; \text{C/Y}=12 ; \text{PMT: END}\)[/tex]



Answer :

GinnyAnswer
To determine which group of values will return the same Present Value (PV) when plugged into the TVM Solver, let's break down the given expression for PV and compare it to the provided options:

The given expression for PV is:
[tex]$ \frac{(\$415) \left((1 + 0.003)^{24} - 1 \right)}{(0.003)(1 + 0.003)^{24}} $[/tex]

From this, we can gather the following details:
- The monthly payment (PMT) is \$415.
- The interest rate per month is [tex]\(0.003\)[/tex] (which is 0.3% monthly).
- The number of periods (N) is 24 months.

Now, we need to identify which option correctly sets up the TVM Solver with these inputs.

### Option Analysis:
Option A:
- [tex]\( N=2 \)[/tex]
- [tex]\( i\% = 3.6 \)[/tex]
- [tex]\( PMT = -415 \)[/tex]
- [tex]\( FV = 0 \)[/tex]
- [tex]\( P/Y = 12 \)[/tex]
- [tex]\( C/Y = 12 \)[/tex]
- [tex]\( PMT: END \)[/tex]

This option has an incorrect number of periods (N=2), and the interest rate is set as an annual rate of 3.6%, which doesn't match our 0.3% monthly rate when converted.

Option B:
- [tex]\( N=2 \)[/tex]
- [tex]\( i\% = 0.3 \)[/tex]
- [tex]\( PMT = -415 \)[/tex]
- [tex]\( FV = 0 \)[/tex]
- [tex]\( P/Y = 12 \)[/tex]
- [tex]\( C/Y = 12 \)[/tex]
- [tex]\( PMT: END \)[/tex]

This option also has an incorrect number of periods (N=2), despite the correct monthly interest rate when interpreted as 0.3%. Thus, this does not match.

Option C:
- [tex]\( N=24 \)[/tex]
- [tex]\( i\% = 0.3 \)[/tex]
- [tex]\( PMT = -415 \)[/tex]
- [tex]\( FV = 0 \)[/tex]
- [tex]\( P/Y = 12 \)[/tex]
- [tex]\( C/Y = 12 \)[/tex]
- [tex]\( PMT: END \)[/tex]

This option matches correctly:
- [tex]\( N=24 \)[/tex] (periods)
- [tex]\( i\% = 0.3 \)[/tex] (monthly interest rate)
- [tex]\( PMT = -415 \)[/tex]
- [tex]\( FV = 0 \)[/tex]
- Payments per Year (P/Y) and Compounding periods per Year (C/Y) are both 12, indicating monthly compounding.

Option D:
- [tex]\( N=24 \)[/tex]
- [tex]\( i\% = 3.6 \)[/tex]
- [tex]\( PMT = -415 \)[/tex]
- [tex]\( FV = 0 \)[/tex]
- [tex]\( P/Y = 12 \)[/tex]
- [tex]\( C/Y = 12 \)[/tex]
- [tex]\( PMT: END \)[/tex]

This option has the correct number of periods (N=24), but the interest rate stated is an annual rate of 3.6%, which would not match the calculated monthly rate when broken down (as monthly it should be 0.3%).

Given our breakdown, the correct option that matches the Present Value expression is:

[tex]\[ \boxed{C} \][/tex]

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