The sum of the two numbers is 12 and their product is 35. What is the sum of the reciprocals of these numbers ?

Options

A. `12/35`
B. `1/35`
C. `35/8`
D. `7/32`

Answer

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

### Step 1: Set up the equations
We are given two pieces of information:
1. The sum of the two numbers:
\[ A + B = 12 \]
2. The product of the two numbers:
\[ A \times B = 35 \]

### Step 2: Express one variable in terms of the other
From the first equation, we can express \( B \) in terms of \( A \):
\[ B = 12 - A \]

### Step 3: Substitute into the product equation
Now, substitute \( B \) into the product equation:
\[ A \times (12 - A) = 35 \]

### Step 4: Expand and rearrange the equation
Expanding the left side gives:
\[ 12A - A^2 = 35 \]
Rearranging this equation leads to:
\[ A^2 - 12A + 35 = 0 \]

### Step 5: Factor the quadratic equation
Now we will factor the quadratic equation:
\[ (A - 5)(A - 7) = 0 \]
This gives us two possible solutions for \( A \):
\[ A = 5 \quad \text{or} \quad A = 7 \]

### Step 6: Find the corresponding values of B
Using \( B = 12 - A \):
- If \( A = 5 \), then \( B = 12 - 5 = 7 \).
- If \( A = 7 \), then \( B = 12 - 7 = 5 \).

Thus, the two numbers are \( 5 \) and \( 7 \).

### Step 7: Calculate the sum of the reciprocals
The sum of the reciprocals of \( A \) and \( B \) is given by:
\[ \frac{1}{A} + \frac{1}{B} = \frac{1}{5} + \frac{1}{7} \]

### Step 8: Find a common denominator and add
The common denominator for \( 5 \) and \( 7 \) is \( 35 \):
\[ \frac{1}{5} = \frac{7}{35}, \quad \frac{1}{7} = \frac{5}{35} \]
Adding these together:
\[ \frac{7}{35} + \frac{5}{35} = \frac{12}{35} \]

### Final Answer
The sum of the reciprocals of the two numbers is:
\[ \frac{12}{35} \]

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