let a,b,c,d,e,f be positive integer such that a/b c/d e/f suppose af - be = -1 - Maths - Probability

Question

let a,b,c,d,e,f be positive integer such that a/b < c/d < e/f
suppose af - be = -1 show that d>=b+f

Neha Sethi · Accepted Answer

Dear student
Given, ab<cd<ef    
(1)And af-be=-1 Since a, b, c, d , e are natural numbers  so  ad, bc , cf and de arealso natural numbers From (1) ad<bc ⇒ad≤bc-1     
(2) cf<de⇒ cf≤de-1       
(3) Equation (2) ×f and (3)×b⇒adf≤bcf-f ; bcf≤bde-btherfore adf≤bde-b-fb+f≤bde-adfb+f≤(be-af)d 
(4)Now af-be=-1therfore be-af=1 by eq (4) b+f≤dd≥b+f
Regards

let a,b,c,d,e,f be positive integer such that a/b < c/d < e/f suppose af - be = -1 .show that d>=b+f