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Question

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VIPSolutions · Accepted Answer

## Objective

**Find the value of $ x $ such that the present worth of cash flows in years  and 9 equals the present worth of an annual series of $300 from years 1 to8, at an interest rate of 10% per year.**

---

## Available Data

- Cash flow in year : $ x $
- Cash flow from year 1 to 8: $300 each year
- Cash flow in year 9: $ x $
- Interest rate, $ i = 10\% $ per year

---

## Step 1: Express the Present Worth (PW) of the Series

### 1. Present Worth of Annual Series ($ P_{A} $)
Formula for present worth of an ordinary annuity:

$$
P = A \frac{1 - (1 + i)^{-n}}{i}
$$

Where:
- $ A = 300 $
- $ i = .10 $
- $ n = 8 $

Plug in values:

$$
P_{A} = 300 \times \frac{1 - (1 + .10)^{-8}}{.10}
$$

$$
(1 + .10)^{-8} = (1.10)^{-8}
$$
$$
(1.10)^{8} = 2.1436
$$
$$
(1.10)^{-8} = \frac{1}{2.1436} = .4665
$$

$$
P_{A} = 300 \times \frac{1 - .4665}{.10} = 300 \times \frac{.5335}{.10} = 300 \times 5.335 = 160.50
$$

---

### 2. Present Worth of $ x $ in Year  and Year 9

$$
PW_{x} = x + \frac{x}{(1.10)^9}
$$

Calculate $ (1.10)^9 $:

$$
(1.10)^9 = 2.35795
$$

So,

$$
PW_{x} = x + \frac{x}{2.35795} = x + .4242x = 1.4242x
$$

---

## Step 2: Set Up Equation and Solve for $ x $

Set the present worths equal to each other:

$$
PW_{A} = PW_{x}
$$
$$
160.50 = 1.4242x
$$

Solve for $ x $:

$$
x = \frac{160.50}{1.4242}
$$

$$
x = 1124.214
$$

---

## Final Answer

**The value of $ x $ is determined to be**  
$$
\boxed{1124.214}
$$
**(rounded to three decimal places).**

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