Question
Rachel deposited $10 in an account earning 5% interest compounded annually.
To the nearest cent, how much interest will she earn in 3 years?
$
Answers
GPT-4o mini
To calculate the interest earned on Rachel's deposit after 3 years with an annual interest rate of 5% compounded annually, we can use the formula for compound interest:
\[
A = P(1 + r)^n
\]
where:
- \(A\) is the amount of money accumulated after n years, including interest.
- \(P\) is the principal amount (initial deposit).
- \(r\) is the annual interest rate (decimal).
- \(n\) is the number of years the money is invested or borrowed.
For Rachel's deposit:
- \(P = 10\) dollars
- \(r = 0.05\) (which is 5% as a decimal)
- \(n = 3\)
Substituting the values into the formula:
\[
A = 10(1 + 0.05)^3
\]
\[
A = 10(1.05)^3
\]
\[
A = 10 \times 1.157625
\]
\[
A \approx 11.57625
\]
Now, we round \(A\) to the nearest cent:
\[
A \approx 11.58
\]
To find the interest earned, we subtract the principal from the total amount:
\[
\text{Interest} = A - P
\]
\[
\text{Interest} = 11.58 - 10
\]
\[
\text{Interest} \approx 1.58
\]
Thus, the interest Rachel will earn in 3 years is approximately **$1.58**.
\[
A = P(1 + r)^n
\]
where:
- \(A\) is the amount of money accumulated after n years, including interest.
- \(P\) is the principal amount (initial deposit).
- \(r\) is the annual interest rate (decimal).
- \(n\) is the number of years the money is invested or borrowed.
For Rachel's deposit:
- \(P = 10\) dollars
- \(r = 0.05\) (which is 5% as a decimal)
- \(n = 3\)
Substituting the values into the formula:
\[
A = 10(1 + 0.05)^3
\]
\[
A = 10(1.05)^3
\]
\[
A = 10 \times 1.157625
\]
\[
A \approx 11.57625
\]
Now, we round \(A\) to the nearest cent:
\[
A \approx 11.58
\]
To find the interest earned, we subtract the principal from the total amount:
\[
\text{Interest} = A - P
\]
\[
\text{Interest} = 11.58 - 10
\]
\[
\text{Interest} \approx 1.58
\]
Thus, the interest Rachel will earn in 3 years is approximately **$1.58**.