Answer :
To calculate how much money you would have in the account after 5 years with a 5.5% interest rate compounded monthly, we can use the compound interest formula:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Where:
- [tex]\( A \)[/tex] is the amount of money accumulated after n years, including interest.
- [tex]\( P \)[/tex] is the principal amount (initial money), which is [tex]$10,000 in this case.
- \( r \) is the annual interest rate (decimal), so 5.5% becomes 0.055.
- \( n \) is the number of times interest is compounded per year. Since it's compounded monthly, \( n \) is 12.
- \( t \) is the number of years the money is invested or borrowed for, which here is 5 years.
Plug in these values into the formula:
\[ A = 10000 \left(1 + \frac{0.055}{12}\right)^{12 \times 5} \]
Now, calculate step-by-step:
1. Calculate the monthly interest rate: \(\frac{0.055}{12} = 0.0045833\).
2. Add 1 to the monthly rate: \(1 + 0.0045833 = 1.0045833\).
3. Calculate the total number of compounding periods: \(12 \times 5 = 60\).
4. Raise the total to the power of 60: \(1.0045833^{60}\).
5. Multiply this result by the principal amount ($[/tex]10,000).
After performing all these calculations, you will find that after 5 years, the amount in the account would be approximately [tex]$13,157.
So, the correct answer is option (a) $[/tex]13,157.
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Where:
- [tex]\( A \)[/tex] is the amount of money accumulated after n years, including interest.
- [tex]\( P \)[/tex] is the principal amount (initial money), which is [tex]$10,000 in this case.
- \( r \) is the annual interest rate (decimal), so 5.5% becomes 0.055.
- \( n \) is the number of times interest is compounded per year. Since it's compounded monthly, \( n \) is 12.
- \( t \) is the number of years the money is invested or borrowed for, which here is 5 years.
Plug in these values into the formula:
\[ A = 10000 \left(1 + \frac{0.055}{12}\right)^{12 \times 5} \]
Now, calculate step-by-step:
1. Calculate the monthly interest rate: \(\frac{0.055}{12} = 0.0045833\).
2. Add 1 to the monthly rate: \(1 + 0.0045833 = 1.0045833\).
3. Calculate the total number of compounding periods: \(12 \times 5 = 60\).
4. Raise the total to the power of 60: \(1.0045833^{60}\).
5. Multiply this result by the principal amount ($[/tex]10,000).
After performing all these calculations, you will find that after 5 years, the amount in the account would be approximately [tex]$13,157.
So, the correct answer is option (a) $[/tex]13,157.
