Bunuel
If x and y are positive integers such that x > y > 1 and z=x/y, which of the following must be true?
I. z > (x − 1)/(y − 1)
II. z < (x − 1)/(y − 1)
III. z > (x + 1)/(y + 1)
A. I only
B. I and II
C. II and III
D. II only
E. I and III
This is a great question to test few properties of a fraction. The properties are -
I) If \(x>y\) and given \(a\) is a positive integer, then
\(\frac{x+a}{y+a}<\frac{x}{y}\) and \(\frac{x−a}{y−a}>\frac{x}{y}\)
II) If \(x<y\) and given that \(a\) a positive integer, then
\(\frac{x+a}{y+a}>\frac{x}{y}\) and \(\frac{x−a}{y−a}<\frac{x}{y}\)
So if you are aware of these properties then you will get the answer directly to the question Option
C-----------------------------------------------------------------
Now, let's use this question to derive the options
Given \(x>y\), multiply both sides by \(1\) (because in the choice given each fraction is increased/decreased by 1)
So we get \(x*1>y*1\), now add \(xy\) to both sides of the inequality
\(=>x+xy>y+xy=>x(y+1)>y(x+1)\)
or \(\frac{x}{y}>\frac{x+1}{y+1} => z>\frac{x+1}{y+1}\). Hence Statement
II must be true
now multiply the inequality \(x>y\) by \(-1\)
\(=>-1*x<-1*y\), now add \(xy\) to both sides of the inequality
\(xy-x<xy-y=>x(y-1)<y(x-1)=>\frac{x}{y}<\frac{x-1}{y-1}=>z<\frac{x-1}{y-1}\). Hence Statement
III must be true
Hi
Bunuel,
May I request to add these properties to the Number Properties article in GC forum or book because this can be very useful for tricky questions. If its already present there, then kindly ignore.