Questions: Use the table to calculate the amount of money that must be invested now at 4% annually, compounded quarterly, to obtain 1,400 in four years. How much money must be invested at 4% annually, compounded quarterly, to obtain 1,400 in four years? (Do not round until the final answer. Then round to the nearest cent as needed.)

Solution Steps

To solve this problem, we need to use the formula for the present value of a future sum of money when interest is compounded quarterly. The formula is:

\[ PV = \frac{FV}{(1 + \frac{r}{n})^{nt}} \]

where:

• \( PV \) is the present value (the amount to be invested now)
• \( FV \) is the future value (\$1,400)
• \( r \) is the annual interest rate (4% or 0.04)
• \( n \) is the number of times the interest is compounded per year (4 for quarterly)
• \( t \) is the number of years (4)

We will plug in the values and calculate \( PV \).

We are given the following values:

• Future Value (\( FV \)): \$1,400
• Annual Interest Rate (\( r \)): 0.04
• Compounding Frequency (\( n \)): 4 (quarterly)
• Time Period (\( t \)): 4 years

To find the present value (\( PV \)), we use the formula:

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

Substituting the known values into the formula:

\[ PV = \frac{1400}{\left(1 + \frac{0.04}{4}\right)^{4 \times 4}} \]

First, we calculate the term inside the parentheses:

\[ 1 + \frac{0.04}{4} = 1 + 0.01 = 1.01 \]

Next, we raise this to the power of \( 16 \) (since \( 4 \times 4 = 16 \)):

\[ (1.01)^{16} \approx 1.1699 \]

Now, we can calculate \( PV \):

\[ PV = \frac{1400}{1.1699} \approx 1193.95 \]

Finally, we round the present value to the nearest cent:

\[ PV \approx 1193.95 \]

Final Answer