If x y z, which of the following can be true?

Question

If  x > y > z , which of the following can be true?

I.  \frac{x}{y} >\frac{y}{z}

II.  \frac{z}{x}>\frac{y}{x}

III.  \frac{z}{y}>\frac{x}{y}

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III

Lampard42 · Answer

I.  \frac{x}{y} >\frac{y}{z}  - Possible

II.  \frac{z}{x}>\frac{y}{x}  - Possible

III.  \frac{z}{y}>\frac{x}{y}  - possible
 
So IMO E

saurabh9gupta · Answer

Hi  Bunuel 
I solved this question and got E but before marking the answer, I saw the options II and III. One has an x in the denominator and one has a y in the denominator.

Can't we cancel them and then reduce 
II. z>y
III. z>x

chetan2u ?

chetan2u · Answer

No, you cannot do this..
 It will depend on the sign of the variables. .

II.  \frac{z}{x}>\frac{y}{x} 
Let z=5, y=3 and x=2......  \frac{5}{2}>\frac{3}{2}  and z>y
But say z=-5, y=3 and x=-2....... \frac{-5}{-2}>\frac{3}{-2}.......\frac{5}{2}>-\frac{3}{2}.  and z<y

Kinshook · Answer

saurabh9gupta If x<0 & y<0 in II & III respectively then the conditions are possible. -2(x)>-3(y)>-4(z) II z/x = 2 y/x = 1.5 z/x > y/x III z/y = 4/3 x/y = 2/3 z/y > x/y

Kinshook · Answer

Asked: If  x > y > z , which of the following can be true?

I.  \frac{x}{y} >\frac{y}{z} 
4(x) > 3(y) > 2(z)
x/y = 4/3
y/x =3/2
x/y>y/z
CAN BE TRUE

II.  \frac{z}{x}>\frac{y}{x} 
-2(x)>-3(y)>-4(z)
z/x = 2
y/x = 1.5
z/x > y/x 
CAN BE TRUE

III.  \frac{z}{y}>\frac{x}{y} 
-2(x)>-3(y)>-4(z)
z/y = 4/3
x/y = 2/3
z/y > x/y 
CAN BE TRUE

IMO E

SatvikVedala · Answer

For Positive values of x,y and z
1) (x/y)>(y/z) is true
Consider x = 6,y = 2,z = 1
For Negative values of x,y and z
2) (z/x)>(y/x) and
3) (z/y)>(x/y)
consider x = -1,y=-2 & z = -6

SatvikVedala · Answer

Another Approach in the form of Number Line

when x,y,z are positive

-inf ------ 1 ------ z ------ y ------ x ------ inf

when x,y,z are negative

-inf ------ z ------ y ------ x ------ 1 ------ inf

Case 1) (x/y)>(y/z)
consider x = 8, y = 4 & z = 2, in this case inequality fails
but consider x = 48, y = 4 & z = 2, in this case inequality works

Case 2) (z/x)>(y/x)
This is not possible when x,y,z are +ve since y>z
so going to the other side of number line
now consider z with higher value z = -48,y = -4, x = -2, in this case inequality works

Similarly for Case 3)(z/y)>(x/y)
consider z with higher value z = -48,y = -4, x = -2, inequality works

if this is a must be True question, i guess all 3 options fail since the cases must be true for
1) x>1
2) 0<x<1
3) x<1

rajdeep212003 · Answer

I don't able to get it can someone plz tell me is i Take x ,y ,z ... 3, 2 ,1 how is possible all should be wrong

Bunuel · Answer

Note that the question asks:  which of the following  can  be true?  NOT   must  be true?

So, while a statement  could  be false for some specific values, it  could  be true for some others. Please review the discussion above for complete solutions.

If x > y > z, which of the following can be true?

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