The following payment plan has a present value of $142,000 : $21,000 today, $16,000 at the end of the first year, $16,000 at the end of the second year, and $X at the end of the third year. The rate of interest is 9.40% per year. How much is X?

We can solve for the unknown value $ X $ by calculating the present value of the payment plan, which includes payments made at different points in time, discounted at an interest rate of 9.40% per year. 

### Step-by-step solution:

1. **Present Value (PV) formula**: 
   \[
   PV = \frac{C_1}{(1 + i)^t} + \frac{C_2}{(1 + i)^t} + \ldots
   \]
   where:
   - $ C_t $ is the cash flow at time $ t $,
   - $ i $ is the interest rate,
   - $ t $ is the time in years.

2. **Given data**:
   - Present value of the entire payment plan: $ PV = 142,000 $
   - First payment today (at time $ t = 0 $): $ 21,000 $
   - Second payment at the end of year 1 (at $ t = 1 $): $ 16,000 $
   - Third payment at the end of year 2 (at $ t = 2 $): $ 16,000 $
   - Fourth payment at the end of year 3 (at $ t = 3 $): $ X $
   - Interest rate: $ 9.40\% $ per year, so $ i = 0.094 $.

3. **Present value of each payment**:
   - Present value of the first payment (today):
     \[
     \frac{21,000}{(1 + 0.094)^0} = 21,000
     \]
   - Present value of the second payment (at $ t = 1 $):
     \[
     \frac{16,000}{(1 + 0.094)^1} = \frac{16,000}{1.094}
     \]
   - Present value of the third payment (at $ t = 2 $):
     \[
     \frac{16,000}{(1 + 0.094)^2} = \frac{16,000}{1.094^2}
     \]
   - Present value of the fourth payment (at $ t = 3 $):
     \[
     \frac{X}{(1 + 0.094)^3} = \frac{X}{1.094^3}
     \]

4. **Sum of present values**:
   According to the problem, the total present value equals $142,000, so:
   \[
   21,000 + \frac{16,000}{1.094} + \frac{16,000}{1.094^2} + \frac{X}{1.094^3} = 142,000
   \]

Let's now solve this equation for $ X $.The value of $ X $, the payment at the end of the third year, is approximately \$121,776.59.

Would you like any further details or have any questions?

Here are 5 related questions to further explore:

1. How would the value of $ X $ change if the interest rate were lower, say 7%?
2. What happens to the present value if an additional payment is added in the 4th year?
3. How does the present value formula apply to continuous compounding of interest?
4. If you want to reduce the present value by 10%, how would the payment amounts be adjusted?
5. Can you solve for $ X $ if the payments occur at mid-year instead of at year-end?

**Tip**: When calculating present values, understanding how future cash flows are discounted is crucial for investment and loan decisions.

The following payment plan has a present value of $142,000: $21,000 today, $16,000 at the end of the first year, $16,000 at the end of the second year, and $X at the end of the third year. The rate of interest is 9.40% per year. How much is X?

This example solves a present value calculation involving multiple cash flows with a 9.40% interest rate. Learn how to discount future payments and find the unknown value X for a payment plan with a present value of $142,000. Suitable for finance students at the undergraduate level.

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