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# If 0 < a < < 1, then which of the following must be true? (A

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If 0 < a < < 1, then which of the following must be true? (A [#permalink]  30 Aug 2018, 07:49
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Question Stats:

92% (01:27) correct 7% (01:02) wrong based on 26 sessions
If $$0 < a < \frac{1}{b}< 1$$, then which of the following must be true?

(A) $$a^2 > a > b > b^2$$
(B) $$b > a > a^2 > b^2$$
(C) $$b^2 > a > a^2 > b$$
(D) $$b^2 > a^2 > b > a$$
(E) $$b^2 > b > a > a^2$$
[Reveal] Spoiler: OA

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Re: If 0 < a < < 1, then which of the following must be true? (A [#permalink]  07 Sep 2018, 20:22
sandy wrote:
If $$0 < a < \frac{1}{b}< 1$$, then which of the following must be true?

(A) $$a^2 > a > b > b^2$$
(B) $$b > a > a^2 > b^2$$
(C) $$b^2 > a > a^2 > b$$
(D) $$b^2 > a^2 > b > a$$
(E) $$b^2 > b > a > a^2$$

Note : both a and 1/b are fraction.

Now, b and 1/b are not same. Here is the tricks.

If b is an fraction 1/b becomes greater that 1. but what? we know 1/b is fraction , it's not greater than 1.

So, b is a integer greater that 1.

Thus ,$$b^2$$ , b are largest and second largest integer.

Now a is fraction. So, a^2 is even smaller than a. If we square a fraction, it become even smaller.

So, sequence: $$b^2>b>a>a^2$$

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Re: If 0 < a < < 1, then which of the following must be true? (A [#permalink]  12 Sep 2018, 07:26
Expert's post
sandy wrote:
If $$0 < a < \frac{1}{b}< 1$$, then which of the following must be true?

(A) $$a^2 > a > b > b^2$$
(B) $$b > a > a^2 > b^2$$
(C) $$b^2 > a > a^2 > b$$
(D) $$b^2 > a^2 > b > a$$
(E) $$b^2 > b > a > a^2$$

Let's solve this by testing values

GIVEN: 0 < a < 1/b < 1
Since a is POSITIVE, we can safely multiply all parts of the inequality by a to get: 0 < a² < a/b < a
So, one thing that is definitely true is that a² < a
(A) a² > a > b > b² NO GOOD. ELIMINATE
(B) b > a > a² > b² WORKS. KEEP
(C) b² > a > a² > b WORKS. KEEP
(D) b² > > b > a NO GOOD. ELIMINATE
(E) b² > b > a > a² WORKS. KEEP

GIVEN: 0 < a < 1/b < 1
Since b is POSITIVE, we can safely multiply all parts of the inequality by b to get: 0 < ab < 1 < b
Hmmm, this new inequality doesn't have b and b² (which is needed to check the answer choices)
So, let's take 0 < ab < 1 < b and multiply all parts by b to get: 0 < ab² < b < b²
Let's check the answer choices to see which ones adhere to the inequality b < b²
(B) b > a > a² > NO GOOD. ELIMINATE
(C) b² > a > a² > b WORKS. KEEP
(E) b² > b > a > a² WORKS. KEEP

We're left with 2 answer choices:
(C) b² > a > a² > b
(E) b² > b > a > a²

Answer choice C says a > b and answer choice E says b > a
So, which is it?

Well, we know that a < 1/b
So, it COULD be the case that a = 1/3 and b = 2, which means b > a

So, the correct answer must be E

Cheers,
Brent
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Re: If 0 < a < < 1, then which of the following must be true? (A [#permalink]  13 Sep 2018, 00:35
Did by plugging values as per inequality:

0< a < 1/b < b --> 0< ab < 1 < b
so ab is a fraction and b is whole or whole + fraction --> took values:
a = 1/3 and b = 2
checking via plug in, only E works --> 4> 2> 1/3 > 1/9

(even if we take b as 1.5 it works).
Re: If 0 < a < < 1, then which of the following must be true? (A   [#permalink] 13 Sep 2018, 00:35
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