M14-33

Question

If  ab 
e 0  and  |a| \lt |b| , which of the following must be negative?

A.  \frac{a}{b} - \frac{b}{a}

B.  \frac{a - b}{a + b}

C.  a^b - b^a

D.   \frac{ab}{a - b}

E.  \frac{b - a}{b}

Bunuel · Accepted Answer

Official Solution:

If  ab 
e 0  and  |a| \lt |b| , which of the following must be negative?

A.  \frac{a}{b} - \frac{b}{a}

B.  \frac{a - b}{a + b}

C.  a^b - b^a

D.   \frac{ab}{a - b}

E.  \frac{b - a}{b}

Squaring the inequality  |a| \lt |b|  results in  a^2 \lt b^2 , which can be reformulated as  a^2 - b^2 \lt 0  or, equivalently,  (a - b)(a + b) \lt 0 . This implies that  a - b  and  a + b  must have opposing signs, hence  \frac{a - b}{a + b}  will always be negative.
 
To discard other options consider  a=-1  and  b=2 .

Answer: B

private · Answer

Bunnel,

option e: (b-a)/b = 1 - (a/b)
Case i: if either a or b is negative , a/b is positive , so the 1 - (a/b) is positive
case ii: if both are of the same sign, since |a|<|b|, 0< a/b < 1 => 1 - (a/b) is always positive
Hence option e.

What am I missing? :shock:

Thanks

Bunuel · Answer

The question asks which of the following must be  negative , not positive.

private · Answer

<img src="{SMILIES_PATH}/bangwall.gif" alt=":wall" title="Bang on the Wall" />

Thanks

alphaseeker · Answer

|a|<|b| is possible in below scenarios; (I don't like using smart numbers, but in that case I could not avoid using them)

a=2 b=3
a=-2 b=-3
a=2 b=-3
a=-2 b=3

If you try each of them individually for each choice, you will get negative result for the equation in choice B.

Senthil7 · Answer

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

Lampard42 · Answer

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

RK007 · Answer

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

yogi02 · Answer

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

Pankaj0901 · Answer

Undertood the answer and agreed. But I have a doubt in the following steps:
Assuming option A is negative
\(\frac{a}{b} - \frac{b}{a}\) < 0
=> a/b < b/a (taking b/a to the other side of the inequality)
=> a^2 < b^2 (cross multiplication) => This inequality is always true since it is given that |a| < |b|..
What am I missing here? As this means that option A must always be < 0 . Which step above is fundamentally wrong here?
Note: I eliminated this option using examples (a, b) such as (4,-1) and (-4,-1) which confirms that the given expression could be negative or positive.

Bunuel · Answer

You cannot cross-multiply a/b < b/a because you don't know the signs of the denominators. If they have the same signs, then yes we get a^2 < b^2 but if they have the opposite sings (one is negative, while another is positive), the we should flip the sign and get a^2 > b^2.

Hope it's clear.

kaladin123 · Answer

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

AnubhavC97 · Answer

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

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.

tanishqgirotra · Answer

Is there an alternate approach to solve this question? Can we solve the question by assuming values?

Bunuel · Answer

"MUST BE TRUE" ("ALWAYS TRUE"/"IS TRUE") questions:
 These questions ask which statement is always true for  every  valid set of numbers. If you can find just one valid set of numbers where a statement is not true, it means the statement is not always true and therefore not the correct answer.

So,  for 'MUST BE TRUE' questions, the plug-in method is good for eliminating an option, but it does not provide a 100% guarantee that an option is always true.

For "MUST BE TRUE" questions, when using the plug-in method, if you find that more than one option appears to be correct for a particular number or set of numbers, try using different numbers to double-check. Reevaluate only those options that were previously considered correct.

"COULD BE TRUE" questions:
 The questions that ask which of the following statements could be true are different.  If you can demonstrate that a statement is true for a specific set of numbers, it implies that the statement could be true and therefore is a correct answer .

You can practice more questions of this type  HERE .

Hope it helps.

phmahi1997 · Answer

I like the solution - it’s helpful.

Premkungg · Answer

I like the solution - it’s helpful.

AbhishekAnil · Answer

I like the solution - it’s helpful. I think this is a high-quality question and I agree with explanation.

M14-33

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