Question
Suppose that S2000 is invested at a rate of 3.1%, compounded quarterly. Assuming that no withdrawals are made, find the total amount after 4 years.
Do not round any intermediate computations, and round your answer to the nearest cent.
Do not round any intermediate computations, and round your answer to the nearest cent.
Answers
GPT-4o mini
To find the total amount after 4 years of investment of S2000 at an interest rate of 3.1% compounded quarterly, we can use the formula for compound interest:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- \(A\) is the amount of money accumulated after n years, including interest.
- \(P\) is the principal amount (the initial amount of money).
- \(r\) is the annual interest rate (decimal).
- \(n\) is the number of times interest is compounded per year.
- \(t\) is the number of years the money is invested for.
Given:
- \(P = 2000\)
- \(r = 3.1\% = 0.031\)
- \(n = 4\) (since it is compounded quarterly)
- \(t = 4\)
Plugging in the values:
\[
A = 2000 \left(1 + \frac{0.031}{4}\right)^{4 \times 4}
\]
Now, calculate the individual components:
1. Calculate the quarterly interest rate:
\[
\frac{0.031}{4} = 0.00775
\]
2. Calculate the total number of compounding periods:
\[
n \times t = 4 \times 4 = 16
\]
3. Substitute these values back into the formula:
\[
A = 2000 \left(1 + 0.00775\right)^{16}
\]
4. Calculate \(1 + 0.00775\):
\[
1 + 0.00775 = 1.00775
\]
5. Raise this to the power of 16:
\[
1.00775^{16} \approx 1.127304
\]
6. Now multiply by the principal:
\[
A \approx 2000 \times 1.127304 \approx 2254.608
\]
7. Finally, round the answer to the nearest cent:
\[
A \approx 2254.61
\]
Thus, the total amount after 4 years is **$2254.61**.
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- \(A\) is the amount of money accumulated after n years, including interest.
- \(P\) is the principal amount (the initial amount of money).
- \(r\) is the annual interest rate (decimal).
- \(n\) is the number of times interest is compounded per year.
- \(t\) is the number of years the money is invested for.
Given:
- \(P = 2000\)
- \(r = 3.1\% = 0.031\)
- \(n = 4\) (since it is compounded quarterly)
- \(t = 4\)
Plugging in the values:
\[
A = 2000 \left(1 + \frac{0.031}{4}\right)^{4 \times 4}
\]
Now, calculate the individual components:
1. Calculate the quarterly interest rate:
\[
\frac{0.031}{4} = 0.00775
\]
2. Calculate the total number of compounding periods:
\[
n \times t = 4 \times 4 = 16
\]
3. Substitute these values back into the formula:
\[
A = 2000 \left(1 + 0.00775\right)^{16}
\]
4. Calculate \(1 + 0.00775\):
\[
1 + 0.00775 = 1.00775
\]
5. Raise this to the power of 16:
\[
1.00775^{16} \approx 1.127304
\]
6. Now multiply by the principal:
\[
A \approx 2000 \times 1.127304 \approx 2254.608
\]
7. Finally, round the answer to the nearest cent:
\[
A \approx 2254.61
\]
Thus, the total amount after 4 years is **$2254.61**.