Kinko store needs to buy new copying machines in 6 years for a total cost of $14000. How much should be deposited now in order to have enough money to pay for the machines? Suppose the interest rate is 10% compounded semi-annually.

To solve this problem, we need to calculate the present value (PV) of $14,000, given that the interest rate is 10% compounded semi-annually, and the machines will be bought in 6 years.

The formula for present value with compound interest is:

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

Where:
- $ PV $ is the present value (the amount that should be deposited now).
- $ FV = 14,000 $ is the future value (the cost of the machines).
- $ r = 0.10 $ is the annual interest rate (10%).
- $ n = 2 $ is the number of times the interest is compounded per year (semi-annually).
- $ t = 6 $ is the number of years.

Now, plug in the values:

\[
PV = \frac{14,000}{(1 + \frac{0.10}{2})^{2 \times 6}}
\]

Let me calculate this for you.The present value is approximately **$7,795.72**. This means Kinko should deposit about **$7,795.72** now in order to have $14,000 in 6 years, with an interest rate of 10% compounded semi-annually.

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:
1. What if the interest was compounded quarterly instead of semi-annually?
2. How does increasing the interest rate affect the present value?
3. What would be the present value if the machines were to be bought in 4 years instead of 6?
4. How would the formula change if the interest were compounded continuously?
5. What is the difference between simple interest and compound interest in this scenario?

**Tip:** Compounding more frequently increases the future value of a deposit, meaning less needs to be deposited initially.

Learn how to calculate the present value of $14,000 for a future purchase of copying machines in 6 years. This problem uses the compound interest formula with a 10% annual interest rate, compounded semi-annually. Suitable for students in Grades 10-12.

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