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Find the present value of the given investment. An investment earns \(10 \%\) per year and is worth \(\$ 5,000\) after 3 months.

Short Answer

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The present value of the given investment is approximately $4,878.13, when considering a 10% annual interest rate and a future value of $5,000 after 3 months.
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Step 1: Recall or provide the formula for present value.

The formula for calculating the present value (PV) of an investment is given by: \[PV = \frac{FV}{(1 + r)^n}\] where \(FV\) is the future value, \(r\) is the interest rate per period, and \(n\) is the number of periods.

Step 2: Transform the interest rate from annual to monthly and convert the time period to months.

In this problem, we are given an annual interest rate of 10%, which needs to be converted into a monthly interest rate. We can do this by dividing the annual interest rate by 12, as there are 12 months in a year. So the monthly interest rate is: \(r=\frac{10\%}{12} \approx 0.008333\) Also, the given investment is worth $5,000 after 3 months, so the number of periods \(n = 3\).

Step 3: Use the present value formula and the values obtained in the previous step to calculate the present value of the given investment.

Now we can plug in our values for \(FV\), \(r\), and \(n\) into the present value formula: \[PV = \frac{5,000}{(1 + 0.008333)^3}\] Calculating the denominator: \((1 + 0.008333)^3 \approx 1.0251\) Now dividing $5,000 by the denominator: \[PV = \frac{5,000}{1.0251} \approx 4,878.13\] The present value of the given investment is approximately $4,878.13.

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