Question
Magan invested $3,200 in an account paying an interest rate of 8, start fraction, 3, divided by, 4, end fraction8
4
3
% compounded quarterly. Vani invested $3,200 in an account paying an interest rate of 8, start fraction, 5, divided by, 8, end fraction8
8
5
% compounded continuously. After 9 years, how much more money would Magan have in his account than Vani, to the nearest dollar?
4
3
% compounded quarterly. Vani invested $3,200 in an account paying an interest rate of 8, start fraction, 5, divided by, 8, end fraction8
8
5
% compounded continuously. After 9 years, how much more money would Magan have in his account than Vani, to the nearest dollar?
Answers
GPT-4o mini
To calculate how much more money Magan will have compared to Vani after 9 years, we will calculate the future value of both investments using the appropriate formulas.
**Magan's Investment (compounded quarterly):**
The formula for compound interest is:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
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 for.
For Magan's account:
- \(P = 3200\)
- \(r = \frac{8 \frac{3}{4}}{100} = 0.0875\) (converting mixed number to decimal: \(8 \frac{3}{4} = 8.75\))
- \(n = 4\) (quarterly compounding)
- \(t = 9\)
Now substitute the values into the formula:
\[
A_M = 3200 \left(1 + \frac{0.0875}{4}\right)^{4 \times 9}
\]
Calculating:
\[
A_M = 3200 \left(1 + \frac{0.0875}{4}\right)^{36}
\]
\[
= 3200 \left(1 + 0.021875\right)^{36}
\]
\[
= 3200 \left(1.021875\right)^{36}
\]
Calculating \( (1.021875)^{36} \):
\[
(1.021875)^{36} \approx 2.0236
\]
Now, calculate \(A_M\):
\[
A_M \approx 3200 \times 2.0236 \approx 6475.52
\]
So, Magan's final amount after 9 years is approximately \(\$6476\).
---
**Vani's Investment (compounded continuously):**
The formula for continuous compounding is:
\[
A = Pe^{rt}
\]
where:
- \(e\) is the base of the natural logarithm (approximately equal to 2.71828).
For Vani's account:
- \(P = 3200\)
- \(r = \frac{8 \frac{5}{8}}{100} = 0.085625\) (converting mixed number to decimal: \(8 \frac{5}{8} = 8.625\))
- \(t = 9\)
Substituting these values into the formula:
\[
A_V = 3200 e^{0.085625 \times 9}
\]
Calculating \(0.085625 \times 9\):
\[
0.085625 \times 9 = 0.770625
\]
Now we find \(e^{0.770625}\):
\[
e^{0.770625} \approx 2.1624
\]
Thus,
\[
A_V \approx 3200 \times 2.1624 \approx 6919.68
\]
So, Vani's final amount after 9 years is approximately \(\$6920\).
---
**Finding the difference:**
Now, we find how much more Magan has compared to Vani:
\[
\text{Difference} = A_M - A_V \approx 6476 - 6920 \approx -444
\]
This means Vani has more money than Magan. The absolute difference is:
\[
|\text{Difference}| \approx 444
\]
Thus, Magan has approximately **$444 less** than Vani after 9 years.
**Magan's Investment (compounded quarterly):**
The formula for compound interest is:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
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 for.
For Magan's account:
- \(P = 3200\)
- \(r = \frac{8 \frac{3}{4}}{100} = 0.0875\) (converting mixed number to decimal: \(8 \frac{3}{4} = 8.75\))
- \(n = 4\) (quarterly compounding)
- \(t = 9\)
Now substitute the values into the formula:
\[
A_M = 3200 \left(1 + \frac{0.0875}{4}\right)^{4 \times 9}
\]
Calculating:
\[
A_M = 3200 \left(1 + \frac{0.0875}{4}\right)^{36}
\]
\[
= 3200 \left(1 + 0.021875\right)^{36}
\]
\[
= 3200 \left(1.021875\right)^{36}
\]
Calculating \( (1.021875)^{36} \):
\[
(1.021875)^{36} \approx 2.0236
\]
Now, calculate \(A_M\):
\[
A_M \approx 3200 \times 2.0236 \approx 6475.52
\]
So, Magan's final amount after 9 years is approximately \(\$6476\).
---
**Vani's Investment (compounded continuously):**
The formula for continuous compounding is:
\[
A = Pe^{rt}
\]
where:
- \(e\) is the base of the natural logarithm (approximately equal to 2.71828).
For Vani's account:
- \(P = 3200\)
- \(r = \frac{8 \frac{5}{8}}{100} = 0.085625\) (converting mixed number to decimal: \(8 \frac{5}{8} = 8.625\))
- \(t = 9\)
Substituting these values into the formula:
\[
A_V = 3200 e^{0.085625 \times 9}
\]
Calculating \(0.085625 \times 9\):
\[
0.085625 \times 9 = 0.770625
\]
Now we find \(e^{0.770625}\):
\[
e^{0.770625} \approx 2.1624
\]
Thus,
\[
A_V \approx 3200 \times 2.1624 \approx 6919.68
\]
So, Vani's final amount after 9 years is approximately \(\$6920\).
---
**Finding the difference:**
Now, we find how much more Magan has compared to Vani:
\[
\text{Difference} = A_M - A_V \approx 6476 - 6920 \approx -444
\]
This means Vani has more money than Magan. The absolute difference is:
\[
|\text{Difference}| \approx 444
\]
Thus, Magan has approximately **$444 less** than Vani after 9 years.
d
This is wrong
GPT-4o mini
Let's recalculate both investments carefully to ensure accurate results.
### Magan's Investment (Compounded Quarterly)
Using the compound interest formula:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- \( P = 3200 \)
- \( r = 8 \frac{3}{4} \% = 8.75 \% = 0.0875 \)
- \( n = 4 \) (quarterly compounding)
- \( t = 9 \) years
Substituting the values into the formula:
\[
A_M = 3200 \left(1 + \frac{0.0875}{4}\right)^{4 \times 9}
\]
Calculating:
\[
A_M = 3200 \left(1 + \frac{0.0875}{4}\right)^{36}
\]
\[
= 3200 \left(1 + 0.021875\right)^{36}
\]
\[
= 3200 \left(1.021875\right)^{36}
\]
Calculating \( (1.021875)^{36} \):
\[
(1.021875)^{36} \approx 2.10958092
\]
Now, calculating \( A_M \):
\[
A_M \approx 3200 \times 2.10958092 \approx 6750.85
\]
So, Magan's final amount after 9 years is approximately **$6751**.
---
### Vani's Investment (Compounded Continuously)
Using the formula for continuous compounding:
\[
A = Pe^{rt}
\]
Where:
- \( P = 3200 \)
- \( r = 8 \frac{5}{8} \% = 8.625 \% = 0.08625 \)
- \( t = 9 \) years
Now we substitute the values:
\[
A_V = 3200 e^{0.08625 \times 9}
\]
Calculating \(0.08625 \times 9\):
\[
0.08625 \times 9 = 0.77625
\]
Now calculating \( e^{0.77625} \):
\[
e^{0.77625} \approx 2.173996
\]
Calculating \( A_V \):
\[
A_V \approx 3200 \times 2.173996 \approx 6943.19
\]
So, Vani's final amount after 9 years is approximately **$6943**.
---
### Finding the Difference
Now, let's find how much more Magan has compared to Vani.
\[
\text{Difference} = A_M - A_V \approx 6751 - 6943 \approx -192
\]
This means Vani has about **$192 more** than Magan after 9 years.
---
### Final Result
To determine how much **more** money Magan would have than Vani, based on this specific setup, Magan actually has less, specifically **$192 less** than Vani. I hope this clears things up!
### Magan's Investment (Compounded Quarterly)
Using the compound interest formula:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- \( P = 3200 \)
- \( r = 8 \frac{3}{4} \% = 8.75 \% = 0.0875 \)
- \( n = 4 \) (quarterly compounding)
- \( t = 9 \) years
Substituting the values into the formula:
\[
A_M = 3200 \left(1 + \frac{0.0875}{4}\right)^{4 \times 9}
\]
Calculating:
\[
A_M = 3200 \left(1 + \frac{0.0875}{4}\right)^{36}
\]
\[
= 3200 \left(1 + 0.021875\right)^{36}
\]
\[
= 3200 \left(1.021875\right)^{36}
\]
Calculating \( (1.021875)^{36} \):
\[
(1.021875)^{36} \approx 2.10958092
\]
Now, calculating \( A_M \):
\[
A_M \approx 3200 \times 2.10958092 \approx 6750.85
\]
So, Magan's final amount after 9 years is approximately **$6751**.
---
### Vani's Investment (Compounded Continuously)
Using the formula for continuous compounding:
\[
A = Pe^{rt}
\]
Where:
- \( P = 3200 \)
- \( r = 8 \frac{5}{8} \% = 8.625 \% = 0.08625 \)
- \( t = 9 \) years
Now we substitute the values:
\[
A_V = 3200 e^{0.08625 \times 9}
\]
Calculating \(0.08625 \times 9\):
\[
0.08625 \times 9 = 0.77625
\]
Now calculating \( e^{0.77625} \):
\[
e^{0.77625} \approx 2.173996
\]
Calculating \( A_V \):
\[
A_V \approx 3200 \times 2.173996 \approx 6943.19
\]
So, Vani's final amount after 9 years is approximately **$6943**.
---
### Finding the Difference
Now, let's find how much more Magan has compared to Vani.
\[
\text{Difference} = A_M - A_V \approx 6751 - 6943 \approx -192
\]
This means Vani has about **$192 more** than Magan after 9 years.
---
### Final Result
To determine how much **more** money Magan would have than Vani, based on this specific setup, Magan actually has less, specifically **$192 less** than Vani. I hope this clears things up!