The difference and the product of two numbers are 5 and 36 respectively. The difference of their reciprocals is

Question

Accepted Answer

To answer the question, let's denote the two numbers as $x$ and $y$.

**Given Information:**
1.  The difference of the two numbers is 5.
    This can be written as: $x - y = 5$ (assuming $x > y$ without loss of generality, as the difference of reciprocals will also be consistent).
2.  The product of the two numbers is 36.
    This can be written as: $x \cdot y = 36$

**What we need to find:**
The difference of their reciprocals.
The reciprocals of $x$ and $y$ are $\frac{1}{x}$ and $\frac{1}{y}$ respectively.
The difference of their reciprocals would be $\frac{1}{y} - \frac{1}{x}$ (to maintain a positive difference, consistent with $x-y=5$).

**Calculation:**
Let's find the expression for the difference of the reciprocals:
$\frac{1}{y} - \frac{1}{x}$

To subtract these fractions, we find a common denominator, which is $xy$:
$\frac{1}{y} - \frac{1}{x} = \frac{1 \cdot x}{y \cdot x} - \frac{1 \cdot y}{x \cdot y} = \frac{x}{xy} - \frac{y}{xy} = \frac{x - y}{xy}$

Now, we substitute the given values into this expression:
We know $x - y = 5$
And we know $xy = 36$

So, the difference of their reciprocals is $\frac{5}{36}$.

**Justification for Option 1 (Correct Option):**

*   **Option 1: $\frac{5}{36}$**
    As calculated above, by setting up the expression for the difference of reciprocals ($\frac{x-y}{xy}$) and substituting the given values for the difference of the numbers ($x-y=5$) and their product ($xy=36$), we directly arrive at $\frac{5}{36}$. This perfectly matches Option 1.

**Justification for Other Options (Incorrect Options):**

*   **Option 2: $\frac{9}{5}$**
    This value is the reciprocal of the correct answer multiplied by some factor, or it would be $\frac{xy}{x-y}$. It does not represent the difference of the reciprocals. If the numbers were, for example, 9 and 4 (difference 5, product 36), then $\frac{1}{4} - \frac{1}{9} = \frac{9-4}{36} = \frac{5}{36}$. This option is incorrect.

*   **Option 3: $\frac{5}{9}$**
    This value is not derived from the given information in the context of finding the difference of reciprocals. It is not $\frac{5}{36}$. This option is incorrect.

*   **Option 4: $\frac{31}{36}$**
    This value does not result from the correct calculation for the difference of reciprocals. It might be related to other operations or combinations of the numbers, but it is not the required difference. For instance, if one were to incorrectly add the reciprocals and then subtract something, or if it were related to $(x+y)^2 = (x-y)^2 + 4xy = 5^2 + 4(36) = 25 + 144 = 169$, so $x+y=13$. But this is not what is asked. This option is incorrect.

Therefore, based on the direct calculation using the given information, Option 1 is the correct answer.

The final answer is $\boxed{\frac{5}{36}}$.