Problem

Magan invested 790 dollars in an account paying an interest rate of 8% compounded quarterly. Angel invested $790 in an account paying an interest rate of 8%
compounded continuously. After 7 years, how much more money would Magan have
in his account than Angel, to the nearest dollar?

Solution

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Text explanation

To solve this problem, we will calculate the future value of Magan's and Angel's investments separately using the formulas for compound interest and continuous compounding, respectively. Then, we'll find the difference between the two.

\[ A = P\left(1 + \frac{r}{n}\right)^{nt} \] $ A = Pe^{rt} $

Answer:

Magan would have approximately \$228 more in his account than Angel after 7 years, to the nearest dollar.

Step by Step Solution:

Step 1: Magan's Account (Compounded Quarterly):

The formula for compound interest is: \[ A = P\left(1 + \frac{r}{n}\right)^{nt} \]

Where:

  • \( A \) = the amount of money accumulated after \( n \) years, including interest.
  • \( P \) = the principal amount (the initial sum of money).
  • \( r \) = annual interest rate (decimal).
  • \( n \) = number of times the interest is compounded per year.
  • \( t \) = the time the money is invested for in years.

Given:

  • \( P = \$790 \)
  • \( r = 0.08 \)
  • \( n = 4 \) (since the interest is compounded quarterly).
  • \( t = 7 \) years.

\[ A_M = 790\left(1 + \frac{0.08}{4}\right)^{4 \cdot 7} \]

\[ A_M = 790\left(1 + 0.02\right)^{28} \]

\[ A_M = 790\left(1.02\right)^{28} \]

\[ A_M = 790 \cdot 2.0398873 \]

\[ A_M = 1611.50 \]

Step 2: Angel's Account (Compounded Continuously):

The formula for continuous compounding is: \[ A = Pe^{rt} \]

Where:

  • \( e \) is the base of the natural logarithm (approximately 2.71828).
  • \( r \), \( t \), and \( P \) have the same meaning as in Magan's formula.

Given:

  • \( P = \$790 \)
  • \( r = 0.08 \)
  • \( t = 7 \) years.

\[ A_A = 790e^{0.08 \cdot 7} \]

\[ A_A = 790e^{0.56} \]

\[ A_A = 790 \cdot 1.751073 \]

\[ A_A = 1383.33 \]

Step 3: Difference:

\[ \text{Difference} = A_M - A_A \]

\[ \text{Difference} = 1611.50 - 1383.33 \]

\[ \text{Difference} = 228.17 \]

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