The sum of two numbers is 7 and their product is 12. What is the sum of their reciprocals?

Options

A. `7//12`
B. `8//15`
C. `7//13`
D. `12//7`

Answer

To solve the problem step by step, let's denote the two numbers as \( x \) and \( y \).

### Step 1: Set up the equations
We know from the problem statement:
1. The sum of the two numbers:
\[ x + y = 7 \]
2. The product of the two numbers:
\[ x \cdot y = 12 \]

### Step 2: Find the sum of the reciprocals
The sum of the reciprocals of the two numbers is given by:
\[ \frac{1}{x} + \frac{1}{y} \]
This can be rewritten using a common denominator:
\[ \frac{1}{x} + \frac{1}{y} = \frac{y + x}{xy} \]
Substituting the values we have:
\[ \frac{x + y}{x \cdot y} \]

### Step 3: Substitute the known values
From our earlier equations, we know:
- \( x + y = 7 \)
- \( x \cdot y = 12 \)

Substituting these values into the equation for the sum of the reciprocals:
\[ \frac{x + y}{x \cdot y} = \frac{7}{12} \]

### Step 4: Conclusion
Thus, the sum of the reciprocals of the two numbers is:
\[ \frac{7}{12} \]

### Final Answer
The answer is \( \frac{7}{12} \).

---