If the difference and the product of two numbers are 5 and 36 respectively, then their reciprocals differ by

Options

A. `5/36`
B. `31/36`
C. `5/9`
D. `9/5`

Answer

To solve the problem step by step, we will denote the two numbers as \( x \) and \( y \). We know the following:

1. The difference of the two numbers is given by:
\[ x - y = 5 \quad \text{(Equation 1)} \]

2. The product of the two numbers is given by:
\[ xy = 36 \quad \text{(Equation 2)} \]

### Step 1: Express \( x \) in terms of \( y \)
From Equation 1, we can express \( x \) in terms of \( y \):
\[ x = y + 5 \]

### Step 2: Substitute \( x \) in Equation 2
Now, we substitute \( x \) in Equation 2:
\[ (y + 5)y = 36 \]

### Step 3: Expand and rearrange the equation
Expanding the equation gives:
\[ y^2 + 5y - 36 = 0 \]

### Step 4: Factor the quadratic equation
Next, we need to factor the quadratic equation:
\[ y^2 + 5y - 36 = (y + 9)(y - 4) = 0 \]

### Step 5: Solve for \( y \)
Setting each factor to zero gives us:
\[ y + 9 = 0 \quad \Rightarrow \quad y = -9 \]
\[ y - 4 = 0 \quad \Rightarrow \quad y = 4 \]

### Step 6: Find corresponding \( x \) values
Now we can find the corresponding \( x \) values for each \( y \):
1. If \( y = 4 \):
\[ x = 4 + 5 = 9 \]
2. If \( y = -9 \):
\[ x = -9 + 5 = -4 \]

### Step 7: Calculate the reciprocals
Now we have two pairs of numbers: (9, 4) and (-4, -9). We will calculate the difference of their reciprocals.

For \( x = 9 \) and \( y = 4 \):
\[ \frac{1}{y} - \frac{1}{x} = \frac{1}{4} - \frac{1}{9} \]

### Step 8: Find a common denominator and simplify
The common denominator for 4 and 9 is 36:
\[ \frac{1}{4} = \frac{9}{36}, \quad \frac{1}{9} = \frac{4}{36} \]
Thus,
\[ \frac{1}{4} - \frac{1}{9} = \frac{9}{36} - \frac{4}{36} = \frac{5}{36} \]

### Final Answer
Therefore, the difference of the reciprocals of the two numbers is:
\[ \frac{5}{36} \]