Question
Suppose you invest $1,000 in a savings account. Which account will grow fastest?
Responses
A an account with simple interestan account with simple interest
B an account with interest compounded annuallyan account with interest compounded annually
C an account with interest compounded monthlyan account with interest compounded monthly
D an account with interest compounded continuously
Responses
A an account with simple interestan account with simple interest
B an account with interest compounded annuallyan account with interest compounded annually
C an account with interest compounded monthlyan account with interest compounded monthly
D an account with interest compounded continuously
Answers
GPT-4o mini
To determine which account will grow fastest, let's evaluate each option based on how interest is applied.
1. **Simple Interest (Option A)**: This method only calculates interest on the principal amount. The formula for simple interest is:
\[
A = P(1 + rt)
\]
where \(A\) is the total amount, \(P\) is the principal, \(r\) is the rate, and \(t\) is the time in years.
2. **Interest Compounded Annually (Option B)**: This method calculates interest on both the principal and the interest that has already been added. The formula is:
\[
A = P(1 + r)^t
\]
3. **Interest Compounded Monthly (Option C)**: This method compounds interest more frequently, which will yield more interest over time. The formula is:
\[
A = P\left(1 + \frac{r}{n}\right)^{nt}
\]
where \(n\) is the number of compounding periods per year (in this case, 12 for monthly).
4. **Interest Compounded Continuously (Option D)**: This is the most frequent compounding method. The formula is:
\[
A = Pe^{rt}
\]
where \(e\) is Euler's number (approximately 2.71828).
In summary, the growth of the investments in each account type increases from simple interest > annually compounded > monthly compounded > continuously compounded. Therefore, the account that will grow the fastest is:
**D: an account with interest compounded continuously.**
1. **Simple Interest (Option A)**: This method only calculates interest on the principal amount. The formula for simple interest is:
\[
A = P(1 + rt)
\]
where \(A\) is the total amount, \(P\) is the principal, \(r\) is the rate, and \(t\) is the time in years.
2. **Interest Compounded Annually (Option B)**: This method calculates interest on both the principal and the interest that has already been added. The formula is:
\[
A = P(1 + r)^t
\]
3. **Interest Compounded Monthly (Option C)**: This method compounds interest more frequently, which will yield more interest over time. The formula is:
\[
A = P\left(1 + \frac{r}{n}\right)^{nt}
\]
where \(n\) is the number of compounding periods per year (in this case, 12 for monthly).
4. **Interest Compounded Continuously (Option D)**: This is the most frequent compounding method. The formula is:
\[
A = Pe^{rt}
\]
where \(e\) is Euler's number (approximately 2.71828).
In summary, the growth of the investments in each account type increases from simple interest > annually compounded > monthly compounded > continuously compounded. Therefore, the account that will grow the fastest is:
**D: an account with interest compounded continuously.**