To determine the present value $P$ that must be invested to have a future value $A$, we can use the formula for simple interest:

$A = P \left( 1 + rt \right)$

Where:

• $A$ is the future value,
• $P$ is the present value (the amount to be invested),
• $r$ is the interest rate per period (expressed as a decimal),
• $t$ is the time in years.

Given:

• $A = 4000$,
• $r = 1.75\% = 0.0175$ (as a decimal),
• $t = 99$ months, which is equivalent to $\frac{99}{12} = 8.25$ years.

Now, rearranging the formula to solve for $P$:

$P = \frac{A}{1 + rt}$

Substitute the known values:

$P = \frac{4000}{1 + (0.0175 \times 8.25)}$

$P = \frac{4000}{1 + 0.144375}$

$P = \frac{4000}{1.144375}$

$P \approx 3500.21$

So, the present value $P$ that must be invested is $3500.21.

Would you like more details or have any questions about the solution?

Here are some related questions you might find interesting:

1. How does the present value change if the interest rate is increased?
2. How can the formula for simple interest be used for different time periods (e.g., quarterly, daily)?
3. What happens to the future value if the time is halved?
4. How would you calculate the future value of a deposit with compound interest instead of simple interest?
5. What is the impact of reducing the interest rate on the amount to be invested?

Tip: When dealing with simple interest, remember that the interest is calculated only on the initial principal, so the formula does not take into account compound growth.