To find the present value $P$ for a given future value $A$ at a simple interest rate $r$ after time $t$, we use the simple interest formula for future value:

$A = P(1 + rt)$

Where:

• $A$ is the future value,
• $P$ is the present value (which we need to find),
• $r$ is the annual interest rate (as a decimal),
• $t$ is the time in years.

We are given:

• $A = 5000$ (the future value),
• $r = 1.65\% = 0.0165$,
• $t = 3 \, \text{months} = \frac{3}{12} = 0.25 \, \text{years}$.

Step-by-Step Calculation

Rearrange the formula to solve for $P$:

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

Now substitute the values:

$P = \frac{5000}{1 + 0.0165 \times 0.25}$

Let's calculate the result.The present value $P$ that must be invested to have a future value of $5000 at a simple interest rate of 1.65% after 3 months is approximately $4979.46.

Would you like more details or have any other questions?

Here are some related questions you might find interesting:

1. How does simple interest differ from compound interest in terms of growth?
2. What would be the future value if the interest rate were compounded monthly?
3. How would the present value change if the time were increased to 6 months?
4. How is the formula for simple interest derived?
5. What happens to the present value if the interest rate doubles?

Tip: Always double-check the time period when calculating interest. For simple interest, it's crucial that the time is expressed in the same unit as the interest rate.