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teal
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M16 q 34 26 Jul 2012, 06:02
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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?

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EvaJager
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M16 q 34 26 Jul 2012, 10:00
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teal
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 \(a = 1\) , \(b = -1\) , \(c = -2\) , \(d = 2\) .
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?
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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 \(\sqrt{x^2}=|x|\).

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