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Math Expert
Joined: 02 Sep 2009
Posts: 52231

Question Stats:
74% (01:17) correct 26% (01:31) wrong based on 100 sessions
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Math Expert
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Re M1634
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16 Sep 2014, 00:00
Official Solution:If \(a\), \(b\), \(c\), and \(d\) are nonzero numbers such that \(\frac{a}{b} = \frac{c}{d}\) and \(\frac{a}{d} = \frac{b}{c}\), which of the following must be true? A. \(a = c\) B. \(b = d\) C. \(a = d\) D. \(b = a\) E. \(b = c\) Given: (i) \(\frac{a}{b} = \frac{c}{d}\) and (ii) \(\frac{a}{d} = \frac{b}{c}\). Now, from (i) \(a=\frac{bc}{d}\), substitute \(a\) in (ii) \(\frac{bc}{d^2}=\frac{b}{c}\). Crossmultiply and reduce by \(b\): \(c^2=d^2\), which means that \(c=d\), but this option is not listed among the answer choices so we should try to get the one which is listed. Again, from (i) \(c=\frac{ad}{b}\), substitute \(c\) in (ii): \(\frac{a}{d}=\frac{b^2}{ad}\). Crossmultiply and reduce by \(d\): \(a^2=b^2\), which means that \(a=b\). Answer: D
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Joined: 21 May 2013
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Re: M1634
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03 Jan 2015, 11:13
Or crossmultiply directly to find:
ad=bc ac=bd
In order for those equalities to be true you can see that c=b or a=b because those letters appear on either side of the equality. There is no c=d in the answers so choose a=b.



Manager
Joined: 24 Nov 2013
Posts: 57

we can also solve by testing numbers a/b = c/d let a/b = 1/2 and c/d = 2/4, a=1, b=2,c=2,d=4
since a/d = b/c, we should check whether 1/4 = 2/2...this is not true
lets try making a=b=1 and d=c=3.. a/b = c/d = 1.
a/d = b/c = 1/3
so we need to have a=b and c=d c=d is not present in the options...hence choose b=a



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hi elimination: ad=bc ac=bd adac=bcbd a(dc)=b(cd) a(dc)=b(dc) a=b, hence D. b=a
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Joined: 07 Dec 2014
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Re M1634
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10 Nov 2015, 21:25
I think this is a highquality question and the explanation isn't clear enough, please elaborate. Hi, i have followed following steps.. let me know if it's correct a/b = c/d ad=bc (1) a/d=b/c ac=bd (2) adding (1) and (2) ad+ac=bc+bd a(d+c)=b(c+d) cancelling (d+c) on both sides a=b or b=a Is this method correct.. as it is mentioned that a,b,c,d are non zero I have divided (c+d) both the sides.. but still I am not getting mod(a) = mod(b) expression. Could you please, if this approach is correct



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Re M1634
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12 Jul 2016, 03:44
I think this is a highquality question and I agree with explanation.



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Re: M1634
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05 Apr 2017, 06:06
The answer must be option D. Cross multiply both the statements to get : 1) ad=bc 2) ac=bd Now do 1/2 to get c=d. Substitute it in 2) to get a=b , i.e. option D
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Joined: 02 Jun 2015
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Location: United States

Re: M1634
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05 Jun 2018, 14:30
This is the way I solve it... a/b = c/d Substitute: a=1 b=1 c=2 d=2 1/1 = 2/2 other words 1=1 a/d = b/c Substitute by the same number above 1/2 = 1/2 Checking the answer choices, the only match is D b = a Hope it helps! Thanks! Ale



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Joined: 19 Feb 2018
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Re: M1634
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31 Oct 2018, 22:04
is cross multiplication allowed? We don't know the sign of any number. Please suggest.



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Re: M1634
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31 Oct 2018, 22:07



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Joined: 19 Feb 2018
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Re: M1634
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31 Oct 2018, 22:26
Thanks for the clarification
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