M16-34

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.  |a| = |c|  
B.  |b| = |d|  
C.  |a| = |d|  
D.  |b| = |a|  
E.  |b| = |c|

Bunuel · Answer

Official Solution:

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.  |a| = |c|  
B.  |b| = |d|  
C.  |a| = |d|  
D.  |b| = |a|  
E.  |b| = |c|

First, cross-multiply each equation to obtain:  ad = bc  and  ac = bd . 
 
Next, multiply these equations together, resulting in:  a^2dc = b^2dc . 
 
Given that the unknowns are non-zero, we can safely cancel out  dc  from both sides, yielding:  a^2 = b^2 .
 
By taking the square root of both sides, we arrive at:  |a| = |b| .

Answer: D

harsh8686 · Answer

is cross multiplication allowed? We don't know the sign of any number. Please suggest.

Bunuel · Answer

We are concerned about the sign when dealing with  inequalities : we should keep the sign if we multiply by a positive value and flip the sign when we multiply by a negative value.

For  equations  we can multiply/divide by a variable regardless of its sign (providing it's not 0) or cross-multiply for that matter.

Hope it's clear.

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.

Mehakgyl · Answer

Hello  Bunuel  , I solved this question in a different manner

a/b= c/d & a/d=b/c

so ad/bc = 1 & ac/bd = 1. Since both are equal to one we can equate the 2 equations

ad/bc = ac/bd

hence d^2 = c^2 
 so |d| = |c|

Could you help me understand what is wrong in this approach?

Bunuel · Answer

Nothing is wrong; that option is simply not available in the choices.

Bunuel · Answer

In the answer choices, we only have two variables, so we should aim to eliminate two from either the first or second equation. Multiplication and division are the operations that can cancel out the unknowns when applied, which should serve as the hint. By the way, we could have multiplied the equations directly from the start without needing to cross-multiply each, and we would still arrive at the same answer. I think these techniques come with practice.

fabianreyes · Answer

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

phmahi1997 · Answer

I like the solution - it’s helpful.

SanchitSanjay · Answer

I like the solution - it’s helpful.

Anish27 · Answer

This is a great question that’s helpful for learning.

nikita.nanda1810 · Answer

Bunuel  Can you help with explaining this with a number and what logically forces |a| = |b| for these equalities to be true?

And when is it okay to multiply equations.

Bunuel · Answer

Let a = 3, b = -3, c = 2, d = -2.

Then a/b = 3/(-3) = -1 and c/d = 2/(-2) = -1, so a/b = c/d.
 Also a/d = 3/(-2) = -1.5 and b/c = (-3)/2 = -1.5, so a/d = b/c.
In this example, |a| = |b| because |3| = |-3|.

What logically forces |a| = |b| is:
 From a/b = c/d you get ad = bc.
 From a/d = b/c you get ac = bd.
Multiply them: (ad)(ac) = (bc)(bd) so a^2cd = b^2cd.
 Since c and d are nonzero, cd is nonzero, so you can cancel cd and get a^2 = b^2, which means a = b or a = -b, hence |a| = |b|.

It is fine to multiply two equations when both equations are true at the same time.

Please review the discussion above for more.

Hope it helps.

atharvadixit · Answer

I did not quite understand the solution. how can we multiple ad = bc and ac = bd? I dont understand the logic applied behind this cross multiplication?

Bunuel · Answer

You can multiply them because if two equalities are true, their products are also true.

M16-34

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