Ann and Tom want to establish a fund for their​ grandson's college education. What lump sum must they deposit at aa 99​% annual interest​ rate, compounded monthlymonthly​, in order to have ​$20 comma 00020,000 in the fund at the end of 1010 ​years?

To solve this problem, we need to determine the present value (the lump sum to be deposited now) that will grow to $20,000 over 10 years at a 9% annual interest rate, compounded monthly.

The formula for the future value of a lump sum with interest compounded periodically is:

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

Where:
- $FV$ is the future value, which is $20,000.
- $PV$ is the present value, which we need to find.
- $r$ is the annual interest rate (0.09 for 9%).
- $n$ is the number of times the interest is compounded per year (12 for monthly compounding).
- $t$ is the number of years the money is invested (10 years).

We need to solve for $PV$:

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

Substituting the given values into the formula:

\[
PV = \frac{20000}{\left(1 + \frac{0.09}{12}\right)^{12 \times 10}}
\]

Let's calculate the present value.The lump sum that Ann and Tom must deposit now to have $20,000 in the fund at the end of 10 years, with a 9% annual interest rate compounded monthly, is approximately **$8,158.75**.

Would you like more details or have any questions?

Here are 5 related questions:
1. How would the required lump sum change if the interest rate were 8% instead of 9%?
2. What would be the future value if they deposited $10,000 instead of the calculated lump sum?
3. How does the compounding frequency affect the lump sum needed for a fixed future value?
4. What is the effect of increasing the investment period to 15 years on the required lump sum?
5. How much interest will be earned over the 10 years with the initial deposit?

**Tip:** The frequency of compounding can significantly impact the final amount of interest earned over time. The more frequent the compounding, the higher the future value.

Learn how to calculate the lump sum Ann and Tom need to deposit now for their grandson's college fund to grow to $20,000 in 10 years at a 9% annual interest rate compounded monthly. Detailed step-by-step solution with key formulas and explanations.

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