??

Ali has $300. He lost 1/3 of it and ends up with $200. To have the $300 again, he needs to win $100 which is ½ of the $200. 1/3 is known as the Losing Fraction (LF) and ½ is known as the Winning Fraction (WF). ½ times 1/3 = 1/6. 1/6 is the product of the LF and WF, known as the PLW. Find the LF and the WF if the PLW is 81/400.


81  =  81  x  1
      =  27  x  3
      =   9   x  9   
(a  and  c)

400 = 400  x   1
       = 200  x   2
       = 100  x   4
       =  80   x   5
       =  50   x   8
       =  40   x  10
       =  25   x  16
(b  and  d)

LF  =  a/b
WF  =  c/d


a/b  =  c/d (1 - a/b)
a d  =  c  (b  -  a)

Using the above factors, trial and error,
a = 9, b = 25, c = 9 and =16

LF = 9/25 and WF = 9/16