Question
Melanie invests $4,000 into an account offering 4% interest compounded annually. Gina invests $4,000 into a simple interest savings account offering a 4.5% interest rate. What is the dollar amount of the greater balance at the end of five years? Round to the nearest dollar. Use the formula A = P (1 + r) Superscript t for compound interest; Use the formula I = P r t for simple interest.
Answers
GPT-4o mini
To compare the balances of Melanie's and Gina's investments after five years, we can use the formulas for compound interest and simple interest as provided.
### Melanie's Investment (Compound Interest)
Melanie's investment uses the formula:
\[
A = P(1 + r)^t
\]
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).
- \(t\) is the time the money is invested for in years.
For Melanie:
- \(P = 4000\)
- \(r = 0.04\) (4% interest)
- \(t = 5\) years
Calculating the amount:
\[
A = 4000(1 + 0.04)^5
\]
\[
A = 4000(1.04)^5
\]
\[
A = 4000 \times 1.216652902 \quad (\text{using a calculator for } 1.04^5 \approx 1.216652902)
\]
\[
A \approx 4866.61
\]
Rounding to the nearest dollar:
\[
A \approx 4867
\]
### Gina's Investment (Simple Interest)
Gina's investment uses the formula:
\[
I = P r t
\]
Where:
- \(I\) is the interest earned.
- \(P\) is the principal amount.
- \(r\) is the annual interest rate (decimal).
- \(t\) is the time the money is invested for in years.
The total amount \(A\) for simple interest can be computed as:
\[
A = P + I = P + P r t = P(1 + rt)
\]
For Gina:
- \(P = 4000\)
- \(r = 0.045\) (4.5% interest)
- \(t = 5\) years
Calculating the interest:
\[
I = 4000 \times 0.045 \times 5
\]
\[
I = 4000 \times 0.225 = 900
\]
Now calculating the total amount:
\[
A = 4000 + 900 = 4900
\]
### Comparison
Now we compare the total amounts:
- Melanie's total amount after 5 years: **$4,867**
- Gina's total amount after 5 years: **$4,900**
Thus, the greater balance at the end of five years is:
\[
\text{Gina's Balance: } \boxed{4900}
\]
### Melanie's Investment (Compound Interest)
Melanie's investment uses the formula:
\[
A = P(1 + r)^t
\]
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).
- \(t\) is the time the money is invested for in years.
For Melanie:
- \(P = 4000\)
- \(r = 0.04\) (4% interest)
- \(t = 5\) years
Calculating the amount:
\[
A = 4000(1 + 0.04)^5
\]
\[
A = 4000(1.04)^5
\]
\[
A = 4000 \times 1.216652902 \quad (\text{using a calculator for } 1.04^5 \approx 1.216652902)
\]
\[
A \approx 4866.61
\]
Rounding to the nearest dollar:
\[
A \approx 4867
\]
### Gina's Investment (Simple Interest)
Gina's investment uses the formula:
\[
I = P r t
\]
Where:
- \(I\) is the interest earned.
- \(P\) is the principal amount.
- \(r\) is the annual interest rate (decimal).
- \(t\) is the time the money is invested for in years.
The total amount \(A\) for simple interest can be computed as:
\[
A = P + I = P + P r t = P(1 + rt)
\]
For Gina:
- \(P = 4000\)
- \(r = 0.045\) (4.5% interest)
- \(t = 5\) years
Calculating the interest:
\[
I = 4000 \times 0.045 \times 5
\]
\[
I = 4000 \times 0.225 = 900
\]
Now calculating the total amount:
\[
A = 4000 + 900 = 4900
\]
### Comparison
Now we compare the total amounts:
- Melanie's total amount after 5 years: **$4,867**
- Gina's total amount after 5 years: **$4,900**
Thus, the greater balance at the end of five years is:
\[
\text{Gina's Balance: } \boxed{4900}
\]