Answer:
1107.80
Explanation:
To calculate the compound interest and the total amount after 1 year and 1 month with the interest compounded quarterly, we'll 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 that interest is compounded per year.
- \( t \) is the time the money is invested for, in years.
Given:
- Principal (\( P \)) = $768
- Annual interest rate (\( r \)) = 100% per annum = 1 (as a decimal)
- Compounded quarterly (\( n \)) = 4 times per year
- Time (\( t \)) = 1 year and 1 month = 1 + 1/12 years = 13/12 years
Now, we plug in the values:
$$ A = 768 \left(1 + \frac{1}{4}\right)^{4 \times \frac{13}{12}} $$
$$ A = 768 \left(1 + 0.25\right)^{\frac{52}{12}} $$
$$ A = 768 \left(1.25\right)^{\frac{52}{12}} $$
To find the compound interest, we subtract the principal from the total amount:
$$ \text{Compound Interest} = A - P $$
Let's calculate the total amount first:
$$ A = 768 \times 1.25^{\frac{52}{12}} $$
Since the calculation involves an exponent that is not a whole number, we'll use a calculator to find the value:
$$ A \approx 768 \times 1.25^{4.3333} $$
$$ A \approx 768 \times 2.4414 $$
$$ A \approx 1875.7952 $$
Rounding to the nearest cent, the total amount \( A \) is approximately **$1875.80**.
Now, we calculate the compound interest:
$$ \text{Compound Interest} = 1875.80 - 768 $$
$$ \text{Compound Interest} \approx 1107.80 $$
So, the compound interest is approximately **$1107.80**, and the total amount after 1 year and 1 month is **$1875.80**. Please note that the actual calculation might slightly vary depending on the precision of the calculator used.