Thank you for your rating!

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?

4.9

(346 votes)

High School Teacher - Tutor for 5 years

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} $

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

**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:**

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

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

**\[ \text{Difference} = 228.17 \]**

CLick to Rate:

The product of a number x and…
Which distance measures 7 uni…
6.24159 rounded to the hundre…
How many pattern block rhombu…
Round 625166.91718 to the nea…
triple the quotient of 2 1/4 …
is 27.64 equivalent to 27 and…
2 Mrs. Sanchez asked her stud…
Of the 22 students in Mr. Jac…
Three times the difference of…
More