The question asks to determine the present value $P$ that must be invested to achieve a future value $A$ using the simple interest formula.

The given values are:

• Future Value $A = 5000$ dollars
• Interest rate $r = 1.65\%$
• Time $t = 9$ months (which we need to convert into years, as interest rates are typically annual).

Formula for Simple Interest

The formula for the future value in simple interest is: $A = P(1 + rt)$ Where:

• $A$ is the future value
• $P$ is the present value (which we need to find)
• $r$ is the interest rate as a decimal (so 1.65% becomes 0.0165)
• $t$ is the time in years (9 months is $\frac{9}{12} = 0.75$ years).

We can rearrange the formula to solve for $P$: $P = \frac{A}{1 + rt}$

Step-by-Step Calculation

Substitute the given values: $P = \frac{5000}{1 + (0.0165 \times 0.75)}$

1. Calculate the product of $r$ and $t$: $0.0165 \times 0.75 = 0.012375$

2. Add 1 to the result: $1 + 0.012375 = 1.012375$

3. Now, divide $A$ by this value: $P = \frac{5000}{1.012375} \approx 4939.13$

So, the present value $P$ that must be invested is approximately $4939.13.

Would you like more details on any part of this solution or have any further questions?

Related Questions:

1. How is simple interest different from compound interest?
2. What is the formula for compound interest and how would the present value change using it?
3. If the interest rate was compounded quarterly, how would the present value change?
4. What happens to the present value if the time increases to 1 year instead of 9 months?
5. How would you calculate the future value if only the present value and time are given?

Tip:

Always ensure time $t$ is in years when using annual interest rates. If given in months, convert it to years by dividing by 12.