Questions: Consumer Mathematics Finding the present value of an investment earning compound interest To help with his child's college fund, Jose needs to invest. Assuming an interest rate of 2.14% compounded monthly, how much would he have to invest to have 78,900 after 12 years? Do not round any intermediate computations, and round your final answer to the nearest dollar. If necessary, refer to the list of financial formulas.

Solution Steps

To find the present value of an investment earning compound interest, we can use the formula for compound interest:

\[ P = \frac{A}{(1 + \frac{r}{n})^{nt}} \]

where:

• \( P \) is the present value (the amount to be invested now),
• \( A \) is the amount of money accumulated after n years, including interest,
• \( r \) is the annual interest rate (decimal),
• \( n \) is the number of times that interest is compounded per year,
• \( t \) is the time the money is invested for in years.

Given:

• \( A = 78,900 \)
• \( r = 2.14\% = 0.0214 \)
• \( n = 12 \) (compounded monthly)
• \( t = 12 \) years

We need to solve for \( P \).

We are given the following values:

• \( A = 78900 \) (the future value)
• \( r = 0.0214 \) (the annual interest rate)
• \( n = 12 \) (the number of compounding periods per year)
• \( t = 12 \) (the number of years)

We will use the present value formula for compound interest:

\[ P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}} \]

Substituting the known values into the formula:

\[ P = \frac{78900}{\left(1 + \frac{0.0214}{12}\right)^{12 \times 12}} \]

Calculating the expression inside the parentheses:

\[ 1 + \frac{0.0214}{12} = 1 + 0.00178333 \approx 1.00178333 \]

Now, raising this to the power of \( 12 \times 12 = 144 \):

\[ (1.00178333)^{144} \approx 1.296682 \]

Now, substituting this back into the present value formula:

\[ P = \frac{78900}{1.296682} \approx 61044.9169 \]

Rounding \( P \) to the nearest dollar gives:

\[ P \approx 61045 \]

Final Answer