As the prize in a​ contest, you are offered 

​$14 comma 00014,000 

now or 

​$24 comma 60024,600 

in 

55 

years. If the money can be invested at 

99​% 

compounded​ annually, which prize will be worth more in 

55 

​years?

Question content area bottom

Part 1

Choose the option that is most profitable after 

55 

years.

accepting the 

$ 14 comma 000$14,000 

prize nownow

accepting the 

$ 24 comma 600$24,600 

prize in 5 years

To determine which prize is more profitable after 5 years, we need to calculate the future value of the $14,000 if it is invested at 9% interest compounded annually, and compare it to the $24,600 offered after 5 years.

### 1. **Future Value of $14,000 Compounded Annually**
The formula for compound interest is:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- $ A $ = the amount of money accumulated after n years, including interest.
- $ P $ = the principal amount ($14,000).
- $ r $ = annual interest rate (9% or 0.09).
- $ n $ = number of times the interest is compounded per year (1 for annually).
- $ t $ = time the money is invested (5 years).

Let's calculate the future value:

\[
A = 14000 \left(1 + \frac{0.09}{1}\right)^{1 \times 5}
\]

### 2. **Comparison to $24,600 After 5 Years**
We'll compare the future value of the $14,000 to the $24,600 prize offered after 5 years to see which is more profitable.### Results:

- **Future value of $14,000 after 5 years**: \$21,540.74
- **Prize in 5 years**: \$24,600

### Conclusion:
Accepting the \$24,600 prize in 5 years is more profitable, as it exceeds the future value of the \$14,000 invested at 9% interest, which would grow to \$21,540.74.

The better option is to accept the **$24,600 prize in 5 years**.

Would you like further details, or do you have any other questions?

### Related Questions:
1. What would the future value be if the interest rate was 10%?
2. How would the comparison change if the prize was available in 10 years instead of 5?
3. What if the initial investment was $16,000 instead of $14,000?
4. How does the decision change if the interest is compounded semi-annually?
5. What if the $24,600 was available in 4 years instead of 5?

### Tip:
When comparing future values, always ensure you account for the compounding effect, as it significantly influences the growth of an investment over time.

Discover which prize is more profitable after 55 years: $14,000 invested now at 9% annual compound interest or $24,600 received in 55 years. This problem explores compound interest concepts and future value calculations.

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