If the difference and product of two numbers are 5 and 36, respectively, what…

Concept

For two nonzero numbers x and y, subtracting their reciprocals requires a common denominator: 1/y - 1/x = (x - y)/(xy).

Thus, when the difference x - y and the product xy are known, the reciprocal difference can be obtained directly from their ratio.

Application

1. Let the two numbers be x and y, with x - y = 5 and xy = 36.

2. Write the reciprocal difference as 1/y - 1/x = (x - y)/(xy).

3. Substitute the given values: (x - y)/(xy) = 5/36.

4. Therefore, the difference between the reciprocals is 5/36.

Cross-check

A factor pair satisfying the conditions is x = 9 and y = 4. Then 1/4 - 1/9 = (9 - 4)/36 = 5/36, confirming the result.