To calculate the total amount in the account after 5 years with an interest rate of 7% compounded semi-annually, we can use the compound interest formula:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
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 that interest is compounded per unit year.
- \(t\) is the time the money is invested or borrowed for, in years.
Plugging in the values:
- \(P = 100\)
- \(r = 0.07\) (7% expressed as a decimal)
- \(n = 2\) (since the interest is compounded semi-annually)
- \(t = 5\)
Now substitute these values into the formula:
\[ A = 100 \left(1 + \frac{0.07}{2}\right)^{2 \times 5} \]
Calculate \( \frac{0.07}{2} \):
\[ \frac{0.07}{2} = 0.035 \]
Now calculate \( 1 + 0.035 \):
\[ 1 + 0.035 = 1.035 \]
Next, calculate the exponent \(2 \times 5\):
\[ 2 \times 5 = 10 \]
Now raise \(1.035\) to the power of \(10\):
\[ 1.035^{10} \approx 1.4106\quad (\text{using a calculator}) \]
Now multiply by the principal amount:
\[ A = 100 \times 1.4106 \approx 141.06 \]
Rounding to the nearest dollar, we have:
\[ A \approx 141 \]
Therefore, the total amount in the account after 5 years will be $141.