M06-16

Question

If  \frac{a}{b} = \frac{x}{y}  and  \frac{a}{y} = \frac{b}{x} , where  a ,  b ,  x , and  y  are non-zero integers, which of the following must be true?
 
I.  \frac{x}{y}=-1 
 
II.  x=y 
 
III.  |x| = |y|

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

Bunuel · Accepted Answer

Official Solution:

If  \frac{a}{b} = \frac{x}{y}  and  \frac{a}{y} = \frac{b}{x} , where  a ,  b ,  x , and  y  are non-zero integers, which of the following must be true?
 
I.  \frac{x}{y}=-1 
 
II.  x=y 
 
III.  |x| = |y|

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

We start by expressing the second equation as  \frac{a}{b} = \frac{y}{x} , so that  x  and  y  are on the same side. Hence, we have  \frac{a}{b} = \frac{x}{y}  and  \frac{a}{b} = \frac{y}{x} . Therefore,  \frac{x}{y}= \frac{y}{x} , which simplifies to  x^2=y^2 . Taking the square root of both sides gives  |x| = |y| . Hence, only option III is always true.
 
To see why options I and II are not always true, consider that if  x  and  y  have the same sign, then from  |x| = |y|  we get  x=y , and in this case, option I is not true. For example, consider  a=b=x=y=1 . Conversely, if  x  and  y  have opposite signs, then from  |x| = |y|  we get  x=-y , and in this case, option II is not true. For example, consider  a=x=1  and  b=y=-1 .

Answer: C

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

BottomJee · Answer

I think this is a  high-quality  question and I agree with explanation.

ashish2023 · Answer

@ Bunuel 
IF we do not take the square roots and keep solving for  x^2 = y^2 ,

For 1) \frac{x}{y = -1 } , so x= -y, then if we put the value of x in prompt, we still get  x^2 = y^2 . So can be True.

For 2) x = y, then it will definitely follow the prompt  x^2 = y^2 , so this can also be True.

Can you tell me where I am thinking wrong ?

Bunuel · Answer

While both x/y = -1 and x = y   could   be true, the question asks  which of the followwing    must    be true?  and those options are not  necessarily  true.

bauyrzhan01 · Answer

a/b=x/y and a/y=b/x. This is given

From above we have:
ay=bx and 
ax=by;

If we add two equations to each other;

ay+ax=bx+by;
a(y+x)=b(x+y); divide each side by (x+y);
a=b

if ax=by and a=b; then x=y. Option 2 is correct.

Please advise why this calculation is wrong

Bunuel · Answer

You cannot divide by x + y because it could be 0 and division by 0 is not allowed. For instance, consider a = 1, b = -1, x = 1, and y = -1. You should have done the following:

a(y + x) = b(x + y)

(x + y)(a - b) = 0

x + y = 0 or a - b = 0.

x + y = 0, implies x = -y.
a - b = 0, implies a = b, which gives x = y.

Hence, either x = -y or x = y. Hence, |x| = |y|.

agrasan · Answer

I like the solution - it’s helpful.

HammeredDeepu97 · Answer

I don’t quite agree with the solution. "To see why options I and II are not always true" - if you are using always in the solution you should also use it in the question as 'always be true'is clear framing of the question than'must be true'.

Must does not capture the essence. If x^2 = y^2 then what all must be true? ( Is it may/can/always/should) ?

Must does not capture the essence. If x^2 = y^2 then what all must be true? ( Is it may/can/always/should) ?

Bunuel · Answer

You are wrong. The phrase “must be true” already means “true in every possible case that fits the given conditions.” So adding “always” in the question is unnecessary, “must be true” and “always true” mean the same thing. I suggest studying more Must/Could Be True questions for better understanding: https://gmatclub.com/forum/problem-solv ... ected_tags =193 Hope it helps.

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