The formula for calculating the present value $P$ with simple interest is:

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

Where:

• $A$ is the future value,
• $P$ is the present value,
• $r$ is the interest rate (in decimal form),
• $t$ is the time period (in years).

Step 1: Convert the given values:

• $A = 6000$,
• $r = 13.5\% = 0.135$,
• $t = 26 \, \text{weeks} = \frac{26}{52} = 0.5 \, \text{years}$.

Step 2: Rearrange the formula to solve for $P$:

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

Substitute the known values into the equation: $P = \frac{6000}{1 + (0.135 \times 0.5)}$ $P = \frac{6000}{1 + 0.0675}$ $P = \frac{6000}{1.0675}$ $P \approx 5611.57$

Final Answer:

The present value $P$ you must invest is approximately $5611.57.

Let me know if you need further details or have any questions!

Here are some follow-up questions you may find interesting:

1. How would the present value change if the interest rate were higher?
2. What is the future value if the investment amount is increased to $10,000 at the same rate and time?
3. How does simple interest compare with compound interest in terms of calculating future value?
4. What happens to the future value if time $t$ is doubled?
5. How does the duration of the investment affect the present value?

Tip: When dealing with simple interest, always convert the interest rate to a decimal before using it in your calculations.