To determine the present value $P$ that must be invested to achieve a future value $A$ with simple interest, 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 you need to invest),
• $r$ is the annual interest rate (expressed as a decimal),
• $t$ is the time in years.

You are given:

• $A = 16,000$,
• $r = 7.5\% = 0.075$,
• $t = 8$ years.

We need to solve for $P$. Rearranging the formula to isolate $P$:

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

Now, substitute the known values into the equation:

$P = \frac{16,000}{1 + 0.075 \times 8}$

$P = \frac{16,000}{1 + 0.6}$

$P = \frac{16,000}{1.6}$

$P = 10,000$

Thus, the present value you must invest is $10,000.

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Would you like to explore how changes in the interest rate or time affect the present value? Here are some related questions:

1. How does increasing the interest rate affect the present value?
2. What happens to the present value if the investment period is shortened?
3. How would you calculate the interest earned in this scenario?
4. What is the formula for calculating the future value in compound interest?
5. How does the simple interest formula differ from the compound interest formula?

Tip: When dealing with simple interest, remember that the interest is calculated only on the initial principal, unlike compound interest where the interest is calculated on both the principal and the accumulated interest.