`1000(1+r/(1,200))^(12)`
The expression above gives the amount of money, in dollars, generated in a year by a $1,000 deposit in a bank account that pays an annual interest rate of r %, compounded monthly. Which of the following expressions shows how much additional money is generated at an interest rate of 5% than at an interest rate of 3% ?

Options

A. `1,000(1+(5-3)/(1,200))^(12)`
B. `1,000(1+(5/3)/(1,200))^(12)`
C. `(1,000(1+5/(1,200))^(12))/(1,000(1+3/(1,200))^(12))`
D. `1,000(1+5/(1,200))^(12)-1,000(1+3/(1,200))^(12)`

Explanation

To solve the problem, we need to calculate the additional money generated by a $1,000 deposit at two different interest rates: 5% and 3%, compounded monthly.

### Step-by-Step Solution:

1. **Identify the formula for compound interest:**
The formula for the amount \( A \) generated by a principal \( P \) at an interest rate \( r \) compounded monthly for one year is given by:
\[ A = P \left(1 + \frac{r}{1200}\right)^{12} \]
Here, \( P = 1000 \).

2. **Calculate the amount generated at 5% interest:**
Substitute \( r = 5 \) into the formula:
\[ A_{5\%} = 1000 \left(1 + \frac{5}{1200}\right)^{12} \]
Simplifying the expression inside the parentheses:
\[ A_{5\%} = 1000 \left(1 + \frac{1}{240}\right)^{12} \]

3. **Calculate the amount generated at 3% interest:**
Substitute \( r = 3 \) into the formula:
\[ A_{3\%} = 1000 \left(1 + \frac{3}{1200}\right)^{12} \]
Simplifying the expression inside the parentheses:
\[ A_{3\%} = 1000 \left(1 + \frac{1}{400}\right)^{12} \]

4. **Find the additional money generated:**
The additional money generated at 5% compared to 3% is given by:
\[ \text{Additional Money} = A_{5\%} - A_{3\%} \]
Substituting the expressions we found:
\[ \text{Additional Money} = 1000 \left(1 + \frac{5}{1200}\right)^{12} - 1000 \left(1 + \frac{3}{1200}\right)^{12} \]
Factoring out the common term:
\[ \text{Additional Money} = 1000 \left(\left(1 + \frac{5}{1200}\right)^{12} - \left(1 + \frac{3}{1200}\right)^{12}\right) \]

5. **Final Expression:**
The final expression that shows how much additional money is generated at an interest rate of 5% compared to 3% is:
\[ 1000 \left(\left(1 + \frac{5}{1200}\right)^{12} - \left(1 + \frac{3}{1200}\right)^{12}\right) \]