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 \( y \).

### Step 1: Set up the equations
According to the problem, when the sum of the two numbers \( (x + y) \) is multiplied by each number separately, we get two products:
1. \( (x + y) \cdot x = 247 \)
2. \( (x + y) \cdot y = 114 \)

### Step 2: Rewrite the equations
From the first equation, we can express it as:
\[ x^2 + xy = 247 \quad \text{(Equation 1)} \]
From the second equation, we can express it as:
\[ xy + y^2 = 114 \quad \text{(Equation 2)} \]

### Step 3: Add the two equations
Now, we will add Equation 1 and Equation 2:
\[ (x^2 + xy) + (xy + y^2) = 247 + 114 \]
This simplifies to:
\[ x^2 + 2xy + y^2 = 361 \]

### Step 4: Recognize the square of a binomial
The left-hand side can be recognized as the expansion of \( (x + y)^2 \):
\[ (x + y)^2 = 361 \]

### Step 5: Take the square root
To find \( x + y \), we take the square root of both sides:
\[ x + y = \sqrt{361} = 19 \]

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

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