Question
When+Martin+was+born+his+father+set+up+a+5,000+college+fund+for+him+with+an+annual+compound+interest+rate+of+7.5 What+is+the+total+value+of+this+account+after+18+years+and+what+was+the+interest+earned+on+the+account+Round+the+answers+to+two+decimal+places.
The final amount of the college fund is $( ) and the amount of
interest+earned+on+the+account+is+( )
The final amount of the college fund is $( ) and the amount of
interest+earned+on+the+account+is+( )
Answers
GPT-4o mini
To find the total value of the college fund after 18 years with an initial investment of $5,000, an annual compound interest rate of 7.5%, 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.
Given:
- \( P = 5000 \)
- \( r = 7.5\% = 0.075 \)
- \( n = 18 \)
Now, substituting the values into the formula:
\[
A = 5000(1 + 0.075)^{18}
\]
Calculating \( (1 + 0.075) \):
\[
1 + 0.075 = 1.075
\]
Next, calculate \( (1.075)^{18} \):
\[
(1.075)^{18} \approx 3.703
\]
Now, substitute this value back into the equation:
\[
A = 5000 \times 3.703 \approx 18515.00
\]
Now we can calculate the interest earned:
\[
\text{Interest Earned} = A - P = 18515.00 - 5000 = 13515.00
\]
So, the total value of the college fund is approximately $18,515.00, and the interest earned is approximately $13,515.00.
The final answers are:
- The final amount of the college fund is $(18515.00)$ and the amount of interest earned on the account is $(13515.00)$.
\[
A = P(1 + r)^n
\]
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.
Given:
- \( P = 5000 \)
- \( r = 7.5\% = 0.075 \)
- \( n = 18 \)
Now, substituting the values into the formula:
\[
A = 5000(1 + 0.075)^{18}
\]
Calculating \( (1 + 0.075) \):
\[
1 + 0.075 = 1.075
\]
Next, calculate \( (1.075)^{18} \):
\[
(1.075)^{18} \approx 3.703
\]
Now, substitute this value back into the equation:
\[
A = 5000 \times 3.703 \approx 18515.00
\]
Now we can calculate the interest earned:
\[
\text{Interest Earned} = A - P = 18515.00 - 5000 = 13515.00
\]
So, the total value of the college fund is approximately $18,515.00, and the interest earned is approximately $13,515.00.
The final answers are:
- The final amount of the college fund is $(18515.00)$ and the amount of interest earned on the account is $(13515.00)$.