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?

Answer 1

This is wrong

Answer 2

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.

Answer 3

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!