The sum of the reciprocals of two consecutive even numbers is 7/24.
Find the numbers.

Let smaller even integer = x
Then larger even integer = x + 2

"Reciprocal of" means "1 over"

 1      1       7
--- + ----- = ---- 
 x     x+2     24

The LCD is 24x(x+2).  Multiply through by that LCD

          1               1                7
24x(x+2)·--- + 24x(x+2)·----- = 24x(x+2)·---- 
          x              x+2              24

Cancel what will cancel

          1               1                7
24x(x+2)·--- + 24x(x+2)·----- = 24x(x+2)·---- 
          x              x+2              24


                 24(x+2) + 24x = x(x+2)·7 

                24x + 48 + 24x = (x²+2x)7

                      48x + 48 = 7x² + 14x

                             0 = 7x² - 34x - 48 
                             
                             0 = (7x + 8)(x - 6)

                Set first factor = 0

                                  7x + 8 = 0
                                      7x = -8
                                       x = -8/7

                Set second factor = 0

                                    x - 6 = 0
                                        x = 6

Discard fraction answer.  Solution: smaller = 6, larger = 6+2 or 8

Edwin
AnlytcPhil@aol.com