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Math Expert V
Joined: 02 Sep 2009
Posts: 58396
If f is a fraction between -1 and 1, which of the following must be tr  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 60% (02:03) correct 40% (02:08) wrong based on 56 sessions

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If f is a fraction between -1 and 1, which of the following must be true?

I. $$f^6 - f^7 < f^4 - f^5$$
II. $$f^6 + f^7 < f^4 + f^5$$
III. $$f^6 < f^4$$

A. I only
B. II only
C. III only
D. I and III only
E. I, II and III

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Senior PS Moderator D
Status: It always seems impossible until it's done.
Joined: 16 Sep 2016
Posts: 737
GMAT 1: 740 Q50 V40 GMAT 2: 770 Q51 V42 Re: If f is a fraction between -1 and 1, which of the following must be tr  [#permalink]

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1
Bunuel wrote:
If f is a fraction between -1 and 1, which of the following must be true?

I. $$f^6 - f^7 < f^4 - f^5$$
II. $$f^6 + f^7 < f^4 + f^5$$
III. $$f^6 < f^4$$

A. I only
B. II only
C. III only
D. I and III only
E. I, II and III

For any fraction between -1 and 1, we know for sure that higher powers means lower absolute values. And even powers ensures positive values whereas negative powers could be positive or negative depending on the fraction.

Keeping this in mind.

I. $$f^6 - f^7 < f^4 - f^5$$

Re-arranging the terms...

$$f^5 - f^7 < f^4 - f^6$$

The RHS is a positive value as f^4 is greater than f^6 and LHS is either a negative value hence the inequality holds when f is negative.

$$f^5( 1-f^2) < f^4( 1-f^2)$$ since $$1-f^2$$ is positive we can cancel it out and hence the inequality holds.

II. $$f^6 + f^7 < f^4 + f^5$$

Re-arrange..

$$f^7 - f^5< f^4 - f^6$$

The RHS is positive where as LHS is negative when f is positive, hence inequality holds.

when f is negative,

$$f^5 ( f^2 -1) < f^4 ( 1- f^2)$$
$$-f^5 (1-f^2) < f^4 ( 1- f^2)$$

This holds.

III. Always holds as increasing powers of f are smaller.

Hence Option (E) is our choice.

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Director  G
Joined: 19 Oct 2013
Posts: 516
Location: Kuwait
GPA: 3.2
WE: Engineering (Real Estate)
Re: If f is a fraction between -1 and 1, which of the following must be tr  [#permalink]

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Bunuel wrote:
If f is a fraction between -1 and 1, which of the following must be true?

I. $$f^6 - f^7 < f^4 - f^5$$
II. $$f^6 + f^7 < f^4 + f^5$$
III. $$f^6 < f^4$$

A. I only
B. II only
C. III only
D. I and III only
E. I, II and III

f^6 ( 1 - f) < f^4 (1 - f) as the power gets larger the value comes smaller. so this holds true regardless of the sign. The (1-f) cancels and we have f^6 < f^4 similar to III

f^6 (1+f) < f^4 ( 1+f) this also holds true in a similar way to the above. The (1+f) cancels and we have f^6 < f^4 similar to III

f^4 ( f^2 - 1) < 0 whatever is inside the bracket is negative so +ve * -ve < 0 is true.

Manager  B
Joined: 28 Jun 2018
Posts: 73
Re: If f is a fraction between -1 and 1, which of the following must be tr  [#permalink]

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Is 0/1 not considered a fraction?
Director  G
Joined: 19 Oct 2013
Posts: 516
Location: Kuwait
GPA: 3.2
WE: Engineering (Real Estate)
Re: If f is a fraction between -1 and 1, which of the following must be tr  [#permalink]

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hibobotamuss 0/any value other than zero = 0

Thus, it is an integer

This comes from the relationship that 0 is a multiple of every number.

0*1 = 0 etc..

Posted from my mobile device Re: If f is a fraction between -1 and 1, which of the following must be tr   [#permalink] 04 Nov 2018, 02:45
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