To calculate the present value $P$ that must be invested to reach the future value $A$ using simple interest, we use the formula:

$A = P(1 + rt)$

Where:

• $A$ is the future value
• $P$ is the present value
• $r$ is the interest rate (as a decimal)
• $t$ is the time in years

Given:

• $A = 4000$ (future value)
• $r = 12.5\% = 0.125$ (interest rate as a decimal)
• $t = 6$ months $= \frac{6}{12} = 0.5$ years

We rearrange the formula to solve for $P$:

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

Now, substituting the given values:

$P = \frac{4000}{1 + (0.125 \times 0.5)}$

Let's calculate that.The present value $P$ that must be invested is approximately $3764.71$.

Would you like further details or have any questions?

Here are some related questions:

1. How does simple interest differ from compound interest in this scenario?
2. How would the present value change if the time period was 1 year instead of 6 months?
3. How would the calculation change if the interest was compounded monthly?
4. What is the formula for finding the future value in simple interest?
5. What happens if the interest rate is doubled but the time remains the same?

Tip: Always convert time to years when dealing with interest rate calculations, as rates are often given on a per-year basis.