The difference between two numbers is 9 and the difference between their squares is 207. The numbers are :

Options

A. 17 and 8
B. 16 and 7
C. 15 and 6
D. 23 and 14

Answer

To solve the problem step by step, we need to find two numbers based on the conditions given.

### Step 1: Define the Variables
Let the two numbers be \( A \) and \( B \), where \( A > B \).

### Step 2: Set Up the Equations
From the problem, we know:
1. The difference between the two numbers is 9:
\[ A - B = 9 \quad \text{(1)} \]
2. The difference between their squares is 207:
\[ A^2 - B^2 = 207 \quad \text{(2)} \]

### Step 3: Use the Difference of Squares
We can use the difference of squares formula, which states that \( A^2 - B^2 = (A - B)(A + B) \). Substituting equation (1) into equation (2):
\[ A^2 - B^2 = (A - B)(A + B) = 9(A + B) = 207 \]

### Step 4: Solve for \( A + B \)
Now, we can solve for \( A + B \):
\[ 9(A + B) = 207 \]
Dividing both sides by 9:
\[ A + B = \frac{207}{9} = 23 \quad \text{(3)} \]

### Step 5: Solve the System of Equations
Now we have a system of two equations:
1. \( A - B = 9 \) (from equation (1))
2. \( A + B = 23 \) (from equation (3))

We can add these two equations:
\[ (A - B) + (A + B) = 9 + 23 \]
This simplifies to:
\[ 2A = 32 \]
Dividing both sides by 2:
\[ A = 16 \]

### Step 6: Find \( B \)
Now that we have \( A \), we can find \( B \) using equation (3):
\[ A + B = 23 \]
Substituting \( A = 16 \):
\[ 16 + B = 23 \]
Subtracting 16 from both sides:
\[ B = 23 - 16 = 7 \]

### Final Answer
The two numbers are \( A = 16 \) and \( B = 7 \).