If the sum of two numbers be multiplied by each number separately, the products so obtained are 247 and 114. The sum of the numbers is

Options

A. 19
B. 20
C. 21
D. 23

Answer

To solve the problem step by step, we will denote the two numbers as \( x \) and \( y \). According to the problem, the sum of the two numbers multiplied by each number separately gives us two products: 247 and 114.

### Step 1: Set up the equations
From the problem statement, we can set up the following equations:
1. \( (x + y) \cdot x = 247 \)
2. \( (x + y) \cdot y = 114 \)

### Step 2: Express the sum \( x + y \)
Let \( S = x + y \). Then we can rewrite the equations as:
1. \( S \cdot x = 247 \) (Equation 1)
2. \( S \cdot y = 114 \) (Equation 2)

### Step 3: Solve for \( x \) and \( y \)
From Equation 1, we can express \( x \) in terms of \( S \):
\[ x = \frac{247}{S} \]
From Equation 2, we can express \( y \) in terms of \( S \):
\[ y = \frac{114}{S} \]

### Step 4: Substitute \( x \) and \( y \) into the sum equation
Since \( S = x + y \), we can substitute the expressions for \( x \) and \( y \):
\[ S = \frac{247}{S} + \frac{114}{S} \]
Multiplying through by \( S \) to eliminate the denominators gives:
\[ S^2 = 247 + 114 \]

### Step 5: Simplify the equation
Now we can simplify the right side:
\[ S^2 = 361 \]

### Step 6: Solve for \( S \)
Taking the square root of both sides, we find:
\[ S = \sqrt{361} = 19 \]

### Conclusion
Thus, the sum of the two numbers \( x + y \) is \( 19 \).