To find the present value $P$ needed to achieve a future value $A$ using simple interest, we can use the formula:

$A = P(1 + rt)$

where:

• $A$ is the future value,
• $P$ is the present value (which we want to find),
• $r$ is the interest rate per time period,
• $t$ is the time in years.

Since we are given:

• $A = 9000$,
• $r = 12\% = 0.12$,
• $t = 6$ months $= \frac{6}{12} = 0.5$ years,

we can rearrange the formula to solve for $P$:

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

Now, substitute the values:

$P = \frac{9000}{1 + (0.12 \times 0.5)}$

Calculating the denominator:

$1 + (0.12 \times 0.5) = 1 + 0.06 = 1.06$

Then,

$P = \frac{9000}{1.06} \approx 8490.57$

So, the present value $P$ that must be invested is approximately $8490.57.

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Would you like more details on the steps?

Here are 5 related questions:

1. How would the present value change if the interest rate increased?
2. What if the investment period changed from 6 months to 1 year?
3. How would compound interest affect the calculation for present value?
4. What if you needed to calculate the future value instead of the present value?
5. How does simple interest differ from compound interest?

Tip: Remember that with simple interest, the interest is not compounded, meaning it is calculated only on the initial principal.