Question

Solve the following problem.
PV equals $ 23 comma 368​; n equals 76​; iequals 0.025​; PMT equals question mark​;

Answer 1

To solve the problem, we need to find the payment amount \( PMT \) when given the present value \( PV \), number of periods \( n \), and interest rate per period \( ie \).

### Given:
- \( PV = 23,368 \)
- \( n = 76 \)
- \( ie = 0.025 \) (which is 2.5%)

### Formula:
The present value of an annuity can be calculated using the following formula:

\[
PV = PMT \times \left( \frac{1 - (1 + ie)^{-n}}{ie} \right)
\]

We can rearrange this formula to solve for \( PMT \):

\[
PMT = \frac{PV} {\left( \frac{1 - (1 + ie)^{-n}}{ie} \right)}
\]

### Step 1: Calculate \( (1 + ie)^{-n} \)

First, we need to calculate \( (1 + ie)^{-n} \):

\[
(1 + ie) = 1 + 0.025 = 1.025
\]

Now calculate \( (1.025)^{-76} \):

\[
(1.025)^{-76} \approx 0.194332
\]

### Step 2: Substitute in the formula

Next, we can substitute back into the formula:

\[
PMT = \frac{23,368}{\left( \frac{1 - 0.194332}{0.025} \right)}
\]

Calculate \( 1 - 0.194332 \):

\[
1 - 0.194332 \approx 0.805668
\]

Now substitute this value back into the equation:

\[
\frac{0.805668}{0.025} \approx 32.22672
\]

### Step 3: Calculate \( PMT \)

Now, substitute this value back to find \( PMT \):

\[
PMT = \frac{23,368}{32.22672} \approx 726.42
\]

Thus, the calculated payment amount \( PMT \) is approximately:

\[
\boxed{726.42}
\]