If a 0, is, is 1/a a/(b^4 + 3) ?

Question

If a ≠ 0, is  \frac{1}{a} > \frac{a}{b^4 +3} ?

(1)  a = b^2

(2)  a^2 = b^4

This comes from the MGMAT advanced quant guide #4-9.

Bunuel · Accepted Answer

If a ≠ 0, is  \frac{1}{a} > \frac{a}{b^4 +3} ?

Notice that we cannot cross-multiply  \frac{1}{a} > \frac{a}{b^4 +3} , since we don't know whether  a>0  or  a<0 , for the first case we should keep the same sign but for the second case we should flip the sign (when multiplying by negative number we should flip the sign of the inequity).

Also notice that we don't have the same issue with  b^4 +3 , since this expression is always positive:  b^4 +3=nonnegative+positive=positive .

(1) a = b^2:

The question becomes:

Is  \frac{1}{b^2} > \frac{b^2}{b^4 +3} ?

Now, since  b^2>0  then here we can safely cross-multiply and the question becomes:

Is  b^4+3>b^4 ?
Is  3>0 ?

Answer to this question is YES. Sufficient.

(2) a^2 = b^4:

The question becomes:

Is  \frac{1}{a} > \frac{a}{a^2 +3} ?

We have the same problem: we cannot cross-multiply since we don't know the sign of  a . If  a=1  then the answer to the question would be YES but if  a=-1  then the answer to the question would be NO. Not sufficient.

Answer: A.

pavanpuneet · Answer

Its clear now! Thanks again for the prompt response Bunuel

grumpytesttaker · Answer

Bunuel never ceases to amaze me. His reasoning is so sharp and his methods so efficient, I bet he conquers every problem in under 30 seconds. I have been following his posts extensively hoping to learn to think like he does. Lifelong fan right here.

rawjetraw · Answer

whoops

! thanks a ton bunuel. Got it. :-D

PerfectScores · Answer

Statement I is sufficient: (1/b^2) > a/(b^4 + 3) We can cross multiply since both the denominators are positive for sure b^4 + 3 > b^2(b^2) B^4 + 3 > B^4 - Well that is true every time Statement II is insufficient a^2 = b^4 If a is and b are positive then the equation will turn to: B^4 + 3 > B^4 In that case YES If a is negative and b is positive then a = -b^2, substituting it we will get: b^4 + 3 < -1 (b^2)(b^2) b^4 + 3 < -(b^4) - That is not true So we have a YES and a NO hence statement II is insufficient Answer is A

MKS · Answer

It's a Yes/No question. From question itself we can deduce that  (B^4+3)  will always be positive.
Rephrasing question ->  \frac{B^4+3}{A}   >   \frac{A}{1} ? 
Note that ,we can not multiply A in this inequality since A might be negative too.

Stmt 1 -
 A = B^2  => A is square of some number (B in this case) so A will always be positive. So multiply with A both side ->
squaring statement 1 at both sides gives us  A^2 = B^4  so we now can multiply with A at both sides as A will always be positive => Sufficient

Stmt 2 -
 A^2 = B^4  => Here A can be negative also as shown below -
 A = B^2  or  A = -B^2  
So this statement is not Sufficient.

Hence A.

shriirah · Answer

Won't option 1 and 2 give the same output? I feel its D

Bunuel · Answer

No. Please review solution  here  carefully.

If a 0, is, is 1/a > a/(b^4 + 3) ?

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