To solve for $ PV $ (Present Value), you typically use the present value formula, which is fundamental in finance. The formula varies slightly depending on whether you're dealing with a single sum or an annuity. Here's a breakdown of both:

### 1. **Present Value of a Single Sum**
If you want to find the present value of a single future sum of money, the formula is:

\[
PV = \frac{FV}{(1 + r)^n}
\]

- $ PV $ = Present Value
- $ FV $ = Future Value
- $ r $ = Interest rate per period (expressed as a decimal)
- $ n $ = Number of periods

### 2. **Present Value of an Annuity**
If you want to find the present value of a series of future payments (an annuity), the formula is:

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

- $ PV $ = Present Value of the annuity
- $ P $ = Payment amount per period
- $ r $ = Interest rate per period (expressed as a decimal)
- $ n $ = Number of periods

### Steps to Solve for $ PV $:
1. **Identify the type of cash flow** (single sum or annuity).
2. **Plug in the values** for $ FV $, $ r $, and $ n $ if solving for a single sum, or $ P $, $ r $, and $ n $ if solving for an annuity.
3. **Perform the calculations** following the formula.

### Example
Suppose you want to calculate the present value of $10,000 to be received 5 years from now, with an interest rate of 6% per year. The calculation would be:

\[
PV = \frac{10,000}{(1 + 0.06)^5} = \frac{10,000}{1.3382} \approx 7,440.94
\]

So, the present value is approximately $7,440.94.

Would you like more details on this, or do you have any other questions?

### Related Questions:
1. How do you solve for the future value $ FV $ if $ PV $ is known?
2. How does the interest rate $ r $ affect the present value $ PV $?
3. How do you calculate the present value for continuous compounding?
4. What is the present value of a perpetuity?
5. How does changing the number of periods $ n $ impact the present value?

### Tip:
When using these formulas, ensure the interest rate and number of periods are consistent. For example, if the interest rate is annual, the number of periods should also be in years.

Understand how to solve for Present Value (PV) using key financial formulas. This guide explains the concepts of present value for both a single sum and an annuity, complete with example calculations. Ideal for undergraduate finance students and advanced high school math learners.

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