Answer :
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:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Where:
- [tex]\( A \)[/tex] is the future value of the investment,
- [tex]\( P \)[/tex] is the principal amount (initial deposit),
- [tex]\( r \)[/tex] is the annual interest rate (expressed as a decimal),
- [tex]\( n \)[/tex] is the number of times interest is compounded per year,
- [tex]\( t \)[/tex] is the number of years the money is invested.
Given:
- [tex]\( P = 15,000 \)[/tex] (initial deposit),
- [tex]\( r = 0.05 \)[/tex] (5% annual interest rate expressed as a decimal),
- [tex]\( n = 4 \)[/tex] (interest compounding quarterly),
- [tex]\( t = 3 \)[/tex] (number of years).
Let's plug these values into the formula:
[tex]\[ A = 15,000 \left(1 + \frac{0.05}{4}\right)^{4 \times 3} \][/tex]
First, calculate the interest rate per quarter:
[tex]\[ \frac{r}{n} = \frac{0.05}{4} = 0.0125 \][/tex]
Next, find how many times the interest is compounded over the 3 years:
[tex]\[ nt = 4 \times 3 = 12 \][/tex]
Now substitute these values back into the formula:
[tex]\[ A = 15,000 \left(1 + 0.0125\right)^{12} \][/tex]
[tex]\[ A = 15,000 \left(1.0125\right)^{12} \][/tex]
Calculate [tex]\( 1.0125^{12} \)[/tex]:
[tex]\[ 1.0125^{12} \approx 1.1616 \][/tex]
Finally, multiply this by the principal amount:
[tex]\[ A = 15,000 \times 1.1616 \][/tex]
[tex]\[ A \approx 17,424 \][/tex]
Joe will have approximately \$17,424 in the account after 3 years if the interest is compounded quarterly.
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Where:
- [tex]\( A \)[/tex] is the future value of the investment,
- [tex]\( P \)[/tex] is the principal amount (initial deposit),
- [tex]\( r \)[/tex] is the annual interest rate (expressed as a decimal),
- [tex]\( n \)[/tex] is the number of times interest is compounded per year,
- [tex]\( t \)[/tex] is the number of years the money is invested.
Given:
- [tex]\( P = 15,000 \)[/tex] (initial deposit),
- [tex]\( r = 0.05 \)[/tex] (5% annual interest rate expressed as a decimal),
- [tex]\( n = 4 \)[/tex] (interest compounding quarterly),
- [tex]\( t = 3 \)[/tex] (number of years).
Let's plug these values into the formula:
[tex]\[ A = 15,000 \left(1 + \frac{0.05}{4}\right)^{4 \times 3} \][/tex]
First, calculate the interest rate per quarter:
[tex]\[ \frac{r}{n} = \frac{0.05}{4} = 0.0125 \][/tex]
Next, find how many times the interest is compounded over the 3 years:
[tex]\[ nt = 4 \times 3 = 12 \][/tex]
Now substitute these values back into the formula:
[tex]\[ A = 15,000 \left(1 + 0.0125\right)^{12} \][/tex]
[tex]\[ A = 15,000 \left(1.0125\right)^{12} \][/tex]
Calculate [tex]\( 1.0125^{12} \)[/tex]:
[tex]\[ 1.0125^{12} \approx 1.1616 \][/tex]
Finally, multiply this by the principal amount:
[tex]\[ A = 15,000 \times 1.1616 \][/tex]
[tex]\[ A \approx 17,424 \][/tex]
Joe will have approximately \$17,424 in the account after 3 years if the interest is compounded quarterly.
To find the future value, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment/loan, including interest
P = the principal investment amount (the initial deposit or loan amount)
r = the annual interest rate (in decimal)
n = the number of times that interest is compounded per unit t
t = the time the money is invested for, in years
Given:
P = $15,000
r = 5% or 0.05
n = 4 (quarterly compounding)
t = 3 years
Plugging the values into the formula:
A = 15000(1 + 0.05/4)^(4*3)
A = 15000(1 + 0.0125)^12
A = 15000(1.0125)^12
A ≈ 15000(1.40492)
A ≈ $21,073.80
So, Joe will have approximately $21,073.80 in 3 years if the interest is compounded quarterly.
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment/loan, including interest
P = the principal investment amount (the initial deposit or loan amount)
r = the annual interest rate (in decimal)
n = the number of times that interest is compounded per unit t
t = the time the money is invested for, in years
Given:
P = $15,000
r = 5% or 0.05
n = 4 (quarterly compounding)
t = 3 years
Plugging the values into the formula:
A = 15000(1 + 0.05/4)^(4*3)
A = 15000(1 + 0.0125)^12
A = 15000(1.0125)^12
A ≈ 15000(1.40492)
A ≈ $21,073.80
So, Joe will have approximately $21,073.80 in 3 years if the interest is compounded quarterly.
