M16 q 34 : Retired Discussions [Locked]
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# M16 q 34

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Manager
Joined: 13 May 2010
Posts: 124
Followers: 0

Kudos [?]: 13 [0], given: 4

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26 Jul 2012, 05:02
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 $$a = 1$$ , $$b = -1$$ , $$c = -2$$ , $$d = 2$$ .

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?
Director
Joined: 22 Mar 2011
Posts: 612
WE: Science (Education)
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Kudos [?]: 910 [0], given: 43

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26 Jul 2012, 09:00
teal wrote:
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 $$\frac{a}{b} = \frac{cb}{dc}$$ , 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 $$a = 1$$ , $$b = -1$$ , $$c = -2$$ , $$d = 2$$ .

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?

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 $$\sqrt{x^2}=|x|$$.
_________________

PhD in Applied Mathematics
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Re: M16 q 34   [#permalink] 26 Jul 2012, 09:00
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# M16 q 34

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