Question
When Martin was born, his father set up a $5,000 college fund for him with an annual compound interest rate of 7.3%.
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.
(2 points)
The final amount of the college fund is $ ________
and the amount of interest earned on the
account is $ _______
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.
(2 points)
The final amount of the college fund is $ ________
and the amount of interest earned on the
account is $ _______
Answers
GPT-4o mini
To calculate the total value of the college fund after 18 years with an initial investment of $5,000 at an annual compound interest rate of 7.3%, we will use the compound interest formula:
\[
A = P(1 + r/n)^{nt}
\]
where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial sum of money).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the number of years the money is invested or borrowed.
In this case:
- \( P = 5000 \)
- \( r = 0.073 \) (7.3% as a decimal)
- \( n = 1 \) (compounded annually)
- \( t = 18 \)
Plugging the values into the formula gives:
\[
A = 5000(1 + 0.073/1)^{(1 \cdot 18)}
\]
\[
A = 5000(1 + 0.073)^{18}
\]
\[
A = 5000(1.073)^{18}
\]
Calculating \( (1.073)^{18} \):
\[
(1.073)^{18} \approx 3.588
\]
Now, substituting it back into the equation:
\[
A \approx 5000 \times 3.588 \approx 17940
\]
Thus, the total amount in the account after 18 years is approximately $17,940.00.
Now, to find the interest earned:
\[
\text{Interest} = A - P
\]
\[
\text{Interest} = 17940 - 5000 = 12940
\]
Putting it all together, we have:
1. The final amount of the college fund is \( \boxed{17940.00} \).
2. The amount of interest earned on the account is \( \boxed{12940.00} \).
\[
A = P(1 + r/n)^{nt}
\]
where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial sum of money).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the number of years the money is invested or borrowed.
In this case:
- \( P = 5000 \)
- \( r = 0.073 \) (7.3% as a decimal)
- \( n = 1 \) (compounded annually)
- \( t = 18 \)
Plugging the values into the formula gives:
\[
A = 5000(1 + 0.073/1)^{(1 \cdot 18)}
\]
\[
A = 5000(1 + 0.073)^{18}
\]
\[
A = 5000(1.073)^{18}
\]
Calculating \( (1.073)^{18} \):
\[
(1.073)^{18} \approx 3.588
\]
Now, substituting it back into the equation:
\[
A \approx 5000 \times 3.588 \approx 17940
\]
Thus, the total amount in the account after 18 years is approximately $17,940.00.
Now, to find the interest earned:
\[
\text{Interest} = A - P
\]
\[
\text{Interest} = 17940 - 5000 = 12940
\]
Putting it all together, we have:
1. The final amount of the college fund is \( \boxed{17940.00} \).
2. The amount of interest earned on the account is \( \boxed{12940.00} \).