The ratio of the sum of the reciprocal of x and y to the product of the reciprocal of x and y is 1 : 3. What is sum of the numbers x and y?

Options

A. `1/3`
B. `1/2`
C. `1`
D. `2`

Answer

To solve the problem, we need to find the sum of the numbers \( x \) and \( y \) given the ratio of the sum of their reciprocals to the product of their reciprocals.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We are given that the ratio of the sum of the reciprocals of \( x \) and \( y \) to the product of the reciprocals of \( x \) and \( y \) is \( 1 : 3 \).

2. **Expressing the Sum of Reciprocals**:
The sum of the reciprocals of \( x \) and \( y \) can be expressed as:
\[ \frac{1}{x} + \frac{1}{y} = \frac{y + x}{xy} \]

3. **Expressing the Product of Reciprocals**:
The product of the reciprocals of \( x \) and \( y \) can be expressed as:
\[ \frac{1}{x} \cdot \frac{1}{y} = \frac{1}{xy} \]

4. **Setting Up the Ratio**:
According to the problem, we set up the ratio:
\[ \frac{\frac{1}{x} + \frac{1}{y}}{\frac{1}{x} \cdot \frac{1}{y}} = \frac{1}{3} \]
Substituting the expressions from steps 2 and 3 into this ratio gives:
\[ \frac{\frac{y + x}{xy}}{\frac{1}{xy}} = \frac{1}{3} \]

5. **Simplifying the Ratio**:
The expression simplifies as follows:
\[ \frac{y + x}{xy} \cdot xy = y + x = \frac{1}{3} \]

6. **Finding the Sum of \( x \) and \( y \)**:
From the above simplification, we find that:
\[ x + y = \frac{1}{3} \]

### Conclusion:
The sum of the numbers \( x \) and \( y \) is \( \frac{1}{3} \).