The sum of two numbers, when multiplied with each of the numbers separately, then the results of the above multiplications are 2418 and 3666, respectively. Find the difference between the numbers.

Options

A. 16
B. 21
C. 19
D. 23

Answer

To solve the problem step-by-step, we can follow these instructions:

### Step 1: Define the Variables
Let the first number be \( A \) and the second number be \( B \).

### Step 2: Set Up the Equations
According to the problem, we have two equations based on the information provided:
1. \( A \times (A + B) = 2418 \) (Equation 1)
2. \( B \times (A + B) = 3666 \) (Equation 2)

### Step 3: Add the Two Equations
Now, we can add both equations:
\[ A \times (A + B) + B \times (A + B) = 2418 + 3666 \]
This simplifies to:
\[ (A + B) \times (A + B) = 6084 \]
Which can be rewritten as:
\[ (A + B)^2 = 6084 \]

### Step 4: Take the Square Root
Now, we take the square root of both sides:
\[ A + B = \sqrt{6084} \]
Calculating the square root gives:
\[ A + B = 78 \]

### Step 5: Substitute Back to Find A
Now we can substitute \( A + B = 78 \) back into Equation 1:
\[ A \times 78 = 2418 \]
Solving for \( A \):
\[ A = \frac{2418}{78} = 31 \]

### Step 6: Find B
Now that we have \( A \), we can find \( B \):
\[ B = 78 - A = 78 - 31 = 47 \]

### Step 7: Find the Difference
Finally, we calculate the difference between the two numbers:
\[ \text{Difference} = B - A = 47 - 31 = 16 \]

### Final Answer
The difference between the two numbers is \( 16 \).

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