To determine the amount of money Joe will have in 3 years if the interest is compounded quarterly, we will use the compound interest formula:
\(\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]\)
Where:
- \(\( A \)\) is the future value of the investment,
- \(\( P \)\) is the principal amount (initial deposit),
- \(\( r \)\) is the annual interest rate (expressed as a decimal),
- \(\( n \)\) is the number of times interest is compounded per year,
- \(\( t \)\) is the number of years the money is invested.
Given:
- \(\( P = 15,000 \)\) (initial deposit),
- \(\( r = 0.05 \)\) (5% annual interest rate expressed as a decimal),
- \(\( n = 4 \)\) (interest compounding quarterly),
- \(\( t = 3 \)\) (number of years).
Let's plug these values into the formula:
\(\[ A = 15,000 \left(1 + \frac{0.05}{4}\right)^{4 \times 3} \]\)
First, calculate the interest rate per quarter:
\(\[ \frac{r}{n} = \frac{0.05}{4} = 0.0125 \]\)
Next, find how many times the interest is compounded over the 3 years:
\(\[ nt = 4 \times 3 = 12 \]\)
Now substitute these values back into the formula:
\(\[ A = 15,000 \left(1 + 0.0125\right)^{12} \]\)
\(\[ A = 15,000 \left(1.0125\right)^{12} \]\)
Calculate \(\( 1.0125^{12} \)\):
\(\[ 1.0125^{12} \approx 1.1616 \]\)
Finally, multiply this by the principal amount:
\(\[ A = 15,000 \times 1.1616 \]\)
\(\[ A \approx 17,424 \]\)
Joe will have approximately \$17,424 in the account after 3 years if the interest is compounded quarterly.
