Magan invested $3,200 in an account paying an interest rate of 8, start fraction, 3, divided by, 4, end fraction8

Question

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

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 \($6476\).

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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 \($6920\).

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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 2

This is wrong

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.

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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.

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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.

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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!