Question
How much interest will an account earn after 4 years if $200 is invested at 5.5% interest compounded annually? List the interest earned at the end of each year.
Enter the amounts as dollars and cents. For example, $5 would be entered as 5.00. Round to the nearest hundredths if needed.
Year 1 interest = $ .
Year 2 interest = $ .
Year 3 interest = $ .
Year 4 interest = $ .
The total compound interest is $
Enter the amounts as dollars and cents. For example, $5 would be entered as 5.00. Round to the nearest hundredths if needed.
Year 1 interest = $ .
Year 2 interest = $ .
Year 3 interest = $ .
Year 4 interest = $ .
The total compound interest is $
Answers
Answered by
GPT-4o mini
To calculate the interest earned on a $200 investment at a 5.5% annual interest rate 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 (the initial amount of money).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of years the money is invested or borrowed.
In this case, \( P = 200 \), \( r = 0.055 \), and \( n = 4 \).
We will calculate the total amount at the end of each year and then find the interest earned each year.
### Yearly Calculations:
#### Year 1:
\[ A_1 = 200(1 + 0.055)^1 = 200(1.055) = 211.00 \]
Interest for Year 1:
\[ \text{Interest}_1 = A_1 - P = 211.00 - 200 = 11.00 \]
#### Year 2:
\[ A_2 = 200(1 + 0.055)^2 = 200(1.055)^2 = 200(1.113025) = 222.61 \]
Interest for Year 2:
\[ \text{Interest}_2 = A_2 - A_1 = 222.61 - 211.00 = 11.61 \]
#### Year 3:
\[ A_3 = 200(1 + 0.055)^3 = 200(1.055)^3 = 200(1.173432875) = 234.69 \]
Interest for Year 3:
\[ \text{Interest}_3 = A_3 - A_2 = 234.69 - 222.61 = 12.08 \]
#### Year 4:
\[ A_4 = 200(1 + 0.055)^4 = 200(1.055)^4 = 200(1.23613867) = 247.23 \]
Interest for Year 4:
\[ \text{Interest}_4 = A_4 - A_3 = 247.23 - 234.69 = 12.54 \]
### Total Compound Interest:
The total compound interest earned after 4 years is:
\[ \text{Total Interest} = \text{Interest}_1 + \text{Interest}_2 + \text{Interest}_3 + \text{Interest}_4 \]
\[ \text{Total Interest} = 11.00 + 11.61 + 12.08 + 12.54 = 47.23 \]
### Summary of Interest Earned:
- Year 1 interest = $11.00
- Year 2 interest = $11.61
- Year 3 interest = $12.08
- Year 4 interest = $12.54
- The total compound interest is $47.23
So the answers are:
- Year 1 interest = $11.00
- Year 2 interest = $11.61
- Year 3 interest = $12.08
- Year 4 interest = $12.54
- The total compound interest is $47.23
\[ A = P(1 + r)^n \]
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 years the money is invested or borrowed.
In this case, \( P = 200 \), \( r = 0.055 \), and \( n = 4 \).
We will calculate the total amount at the end of each year and then find the interest earned each year.
### Yearly Calculations:
#### Year 1:
\[ A_1 = 200(1 + 0.055)^1 = 200(1.055) = 211.00 \]
Interest for Year 1:
\[ \text{Interest}_1 = A_1 - P = 211.00 - 200 = 11.00 \]
#### Year 2:
\[ A_2 = 200(1 + 0.055)^2 = 200(1.055)^2 = 200(1.113025) = 222.61 \]
Interest for Year 2:
\[ \text{Interest}_2 = A_2 - A_1 = 222.61 - 211.00 = 11.61 \]
#### Year 3:
\[ A_3 = 200(1 + 0.055)^3 = 200(1.055)^3 = 200(1.173432875) = 234.69 \]
Interest for Year 3:
\[ \text{Interest}_3 = A_3 - A_2 = 234.69 - 222.61 = 12.08 \]
#### Year 4:
\[ A_4 = 200(1 + 0.055)^4 = 200(1.055)^4 = 200(1.23613867) = 247.23 \]
Interest for Year 4:
\[ \text{Interest}_4 = A_4 - A_3 = 247.23 - 234.69 = 12.54 \]
### Total Compound Interest:
The total compound interest earned after 4 years is:
\[ \text{Total Interest} = \text{Interest}_1 + \text{Interest}_2 + \text{Interest}_3 + \text{Interest}_4 \]
\[ \text{Total Interest} = 11.00 + 11.61 + 12.08 + 12.54 = 47.23 \]
### Summary of Interest Earned:
- Year 1 interest = $11.00
- Year 2 interest = $11.61
- Year 3 interest = $12.08
- Year 4 interest = $12.54
- The total compound interest is $47.23
So the answers are:
- Year 1 interest = $11.00
- Year 2 interest = $11.61
- Year 3 interest = $12.08
- Year 4 interest = $12.54
- The total compound interest is $47.23