Questions: Solve the problem. Suppose 14,000 is invested with 3.5% interest for 9 years compounded quarterly. What is the resulting value? 19,157.36 19,182.75 20,005.98 21,233.65

Solution Steps

To solve this problem, we need to use the formula for compound interest, which is given by:

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

where:

• \( A \) is the amount of money accumulated after n years, including interest.
• \( P \) is the principal amount (initial investment).
• \( r \) 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.

In this case, \( P = 14000 \), \( r = 0.035 \), \( n = 4 \) (since the interest is compounded quarterly), and \( t = 9 \).

We are given the following values:

• Principal amount \( P = 14000 \)
• Annual interest rate \( r = 0.035 \)
• Number of times interest is compounded per year \( n = 4 \)
• Time in years \( t = 9 \)

The formula for compound interest is given by:

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

Substituting the known values into the formula:

\[ A = 14000 \left(1 + \frac{0.035}{4}\right)^{4 \times 9} \]

Calculating the expression step-by-step:

1. Calculate \( \frac{r}{n} = \frac{0.035}{4} = 0.00875 \).
2. Calculate \( 1 + 0.00875 = 1.00875 \).
3. Calculate \( nt = 4 \times 9 = 36 \).
4. Raise \( 1.00875 \) to the power of \( 36 \):

\[ 1.00875^{36} \approx 1.348850 \]

5. Finally, multiply by \( P \):

\[ A \approx 14000 \times 1.348850 \approx 19157.36 \]

Final Answer