M16 q 34

Question

If  a  ,  b  ,  c  , and  d  are non-zero numbers such that  \frac{a}{b} = \frac{c}{d}  and  \frac{a}{d} = \frac{b}{c}  , which of the following must be true?

|a| = |c| 
  |b| = |d| 
  |a| = |d| 
  |b| = |a| 
  |b| = |c|

Because  \frac{a}{b} = \frac{c}{d}  , it is true that  ad = bc  or  c = a*\frac{d}{b}  . Because  \frac{a}{d} = \frac{b}{c}  , it is true that  ac = bd  or  c = b*\frac{d}{a}  . Thus,  c = a*\frac{d}{b} = b*\frac{d}{a}  . Because  d  is not 0,  \frac{a}{b} = \frac{b}{a}  or  a^2 = b^2  or  |a| = |b|  . To see that the other choices are not necessarily true consider  a = 1  ,  b = -1  ,  c = -2  ,  d = 2  .
The correct answer is D.

Can someone please explain me how to solve this problem efficiently? I don't understand the approach used in the solution to manipulate the variables? I did some variable manipulations but figured that if you don't do that a certain way you end up with a different answer like Mod(c) = Mod(d) which is not one of the answer choices?

EvaJager · Answer

The four numbers can be any non-zero real numbers. You can also use some basic algebra and manipulate the two proportions. For example:

Multiply side-by-side the two equalities. It is in fact just simple multiplication of two fractions. 
You get  \frac{a^2}{bd}=\frac{cb}{dc} , from which it follows that  a^2=b^2  (after reduction and cross-multiplication).
Now, take the square root of both sides, and get  |a|=|b| .

You can do all the manipulation above as all the numbers are non-zero. And don't forget that  \sqrt{x^2}=|x| .

M16 q 34

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