The sum of two numbers is 18 and the difference of their squares is 108. The difference between the numbers is

Options

A. 2
B. 12
C. 6
D. 8

Answer

To solve the problem step by step, we will define the two numbers and use the information given in the question to find the difference between them.

### Step 1: Define the variables
Let the two numbers be \( A \) and \( B \).

### Step 2: Set up the equations
From the problem, we know:
1. The sum of the two numbers is 18:
\[ A + B = 18 \quad \text{(Equation 1)} \]
2. The difference of their squares is 108:
\[ A^2 - B^2 = 108 \quad \text{(Equation 2)} \]

### Step 3: Use the difference of squares formula
We can use the algebraic identity for the difference of squares:
\[ A^2 - B^2 = (A + B)(A - B) \]
Substituting Equation 1 into this identity gives us:
\[ A^2 - B^2 = (A + B)(A - B) = 18(A - B) \]
Thus, we can rewrite Equation 2 as:
\[ 18(A - B) = 108 \]

### Step 4: Solve for \( A - B \)
To isolate \( A - B \), divide both sides of the equation by 18:
\[ A - B = \frac{108}{18} \]
Calculating the right side:
\[ A - B = 6 \]

### Conclusion
The difference between the two numbers \( A \) and \( B \) is:
\[ \text{Difference} = 6 \]