Answer :
To determine how much an investment of $14,900 will be worth in 17 years when it is compounded quarterly at an annual interest rate of 2.7%, we use the compound interest formula.
The compound interest formula is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Where:
- \( A \) is the amount of money accumulated after \( t \) years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (decimal form).
- \( n \) is the number of times the interest is compounded per year.
- \( t \) is the number of years the money is invested or borrowed for.
For this problem:
- \( P = 14,900 \) (the initial investment)
- \( r = 0.027 \) (2.7% annual interest rate, in decimal form)
- \( n = 4 \) (since the interest is compounded quarterly)
- \( t = 17 \) years
Next, we plug these values into the compound interest formula:
[tex]\[ A = 14,900 \left(1 + \frac{0.027}{4}\right)^{4 \times 17} \][/tex]
Step-by-step:
1. Calculate the quarterly interest rate:
[tex]\[ \frac{0.027}{4} = 0.00675 \][/tex]
2. Add 1 to the quarterly interest rate:
[tex]\[ 1 + 0.00675 = 1.00675 \][/tex]
3. Determine the total number of compounding periods over the 17 years:
[tex]\[ 4 \times 17 = 68 \][/tex]
4. Raise the base (1.00675) to the power of the total number of compounding periods (68):
[tex]\[ 1.00675^{68} \][/tex]
5. Multiply this result by the principal amount ($14,900):
[tex]\[ A = 14,900 \times 1.00675^{68} \approx 23,542.78 \][/tex]
So, after 17 years, the investment will be worth approximately $23,542.78.
Therefore, the correct answer is:
[tex]\[ \text{B. } 23,542.78 \][/tex]
The compound interest formula is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Where:
- \( A \) is the amount of money accumulated after \( t \) years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (decimal form).
- \( n \) is the number of times the interest is compounded per year.
- \( t \) is the number of years the money is invested or borrowed for.
For this problem:
- \( P = 14,900 \) (the initial investment)
- \( r = 0.027 \) (2.7% annual interest rate, in decimal form)
- \( n = 4 \) (since the interest is compounded quarterly)
- \( t = 17 \) years
Next, we plug these values into the compound interest formula:
[tex]\[ A = 14,900 \left(1 + \frac{0.027}{4}\right)^{4 \times 17} \][/tex]
Step-by-step:
1. Calculate the quarterly interest rate:
[tex]\[ \frac{0.027}{4} = 0.00675 \][/tex]
2. Add 1 to the quarterly interest rate:
[tex]\[ 1 + 0.00675 = 1.00675 \][/tex]
3. Determine the total number of compounding periods over the 17 years:
[tex]\[ 4 \times 17 = 68 \][/tex]
4. Raise the base (1.00675) to the power of the total number of compounding periods (68):
[tex]\[ 1.00675^{68} \][/tex]
5. Multiply this result by the principal amount ($14,900):
[tex]\[ A = 14,900 \times 1.00675^{68} \approx 23,542.78 \][/tex]
So, after 17 years, the investment will be worth approximately $23,542.78.
Therefore, the correct answer is:
[tex]\[ \text{B. } 23,542.78 \][/tex]
