A bank offers a 7% interest rate compounded two times per year on new savings deposits. If a customer deposits $100.00 in the account, how much will be in the account in total after 5 years? Round to the nearest dollar.(1 point)
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Answers
Answered by
GPT-4o mini
To calculate the total amount in the account after 5 years with a 7% interest rate compounded semi-annually, 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 deposit or investment).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the number of years the money is invested or borrowed.
Given:
- \( P = 100.00 \)
- \( r = 0.07 \) (7%)
- \( n = 2 \) (compounded twice a year)
- \( t = 5 \)
Plugging in the values:
\[
A = 100 \left(1 + \frac{0.07}{2}\right)^{2 \times 5}
\]
Calculating step by step:
1. Calculate \( \frac{r}{n} \):
\[
\frac{0.07}{2} = 0.035
\]
2. Calculate \( nt \):
\[
2 \times 5 = 10
\]
3. Now substitute these values back into the formula:
\[
A = 100 \left(1 + 0.035\right)^{10}
\]
4. Calculate \( 1 + 0.035 \):
\[
1 + 0.035 = 1.035
\]
5. Now raise \( 1.035 \) to the power of 10:
\[
1.035^{10} \approx 1.4107
\]
6. Finally, multiply by the principal:
\[
A \approx 100 \times 1.4107 \approx 141.07
\]
Rounding to the nearest dollar:
\[
A \approx 141
\]
Therefore, the total amount in the account after 5 years will be approximately **$141.00**.
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial deposit or investment).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the number of years the money is invested or borrowed.
Given:
- \( P = 100.00 \)
- \( r = 0.07 \) (7%)
- \( n = 2 \) (compounded twice a year)
- \( t = 5 \)
Plugging in the values:
\[
A = 100 \left(1 + \frac{0.07}{2}\right)^{2 \times 5}
\]
Calculating step by step:
1. Calculate \( \frac{r}{n} \):
\[
\frac{0.07}{2} = 0.035
\]
2. Calculate \( nt \):
\[
2 \times 5 = 10
\]
3. Now substitute these values back into the formula:
\[
A = 100 \left(1 + 0.035\right)^{10}
\]
4. Calculate \( 1 + 0.035 \):
\[
1 + 0.035 = 1.035
\]
5. Now raise \( 1.035 \) to the power of 10:
\[
1.035^{10} \approx 1.4107
\]
6. Finally, multiply by the principal:
\[
A \approx 100 \times 1.4107 \approx 141.07
\]
Rounding to the nearest dollar:
\[
A \approx 141
\]
Therefore, the total amount in the account after 5 years will be approximately **$141.00**.