The difference between a number and its reciprocal is 5/6. Find all possible numbers.

Question

World Problem

Answer 1

Let x = the number(s).  Its reciprocal is 1/x:

 

x - (1/x) = 5/6

 

Multiply both sides by x:

 

x² - 1 = 5x/6

 

Multiply both sides by 6:

 

6x² - 6 = 5x

 

Subtract 5x from both sides:

 

6x² - 5x - 6 = 0

 

Factors to:

 

(2x-3)(3x+2) = 0

 

x = 3/2 and -2/3

 

Check:

3/2 - 2/3 = 5/6

9/6 - 4/6 = 5/6

5/6 = 5/6    CHECK!

 

-2/3 - (-3/2) = 5/6

-2/3 + 3/2 = 5/6

-4/6 + 9/6 = 5/6

5/6 = 5/6   CHECK!