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# If 0 < a < 1, which of the following is the greatest?

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If 0 < a < 1, which of the following is the greatest? [#permalink]

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10 Sep 2017, 04:13
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If 0 < a < 1, which of the following is the greatest?

A. $$(-2a)^{(-2)}$$

B. $$\frac{1}{a^{(-2)}}$$

C. $$(\frac{1}{a})^{(-2)}$$

D. $$a^{(-2)}$$

E. $$a^2$$
[Reveal] Spoiler: OA

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If 0 < a < 1, which of the following is the greatest? [#permalink]

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10 Sep 2017, 11:55
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Bunuel wrote:
If 0 < a < 1, which of the following is the greatest?

A. $$(-2a)^{(-2)}$$

B. $$\frac{1}{a^{(-2)}}$$

C. $$(\frac{1}{a})^{(-2)}$$

D. $$a^{(-2)}$$

E. $$a^2$$

as $$a$$ is positive and $$a<1$$, so $$\frac{1}{a}>1$$. Hence a fraction where $$a$$ is in the denominator and numerator is $$1$$ will have a value greater than $$1$$. Hence we can straight away eliminate options B, C & E as $$a$$ is in the numerator (essentially option B, C & E are same $$= a^2$$ and as it is given $$a<1$$, so squaring both sides will yield $$a^2<1$$)

Option A: can be written as $$\frac{1}{(2a)^{2}}$$ $$= \frac{1}{4a^2}$$

Option D: can be written as $$\frac{1}{a^2}$$. Multiply the numerator and the denominator by $$4$$. we get
$$\frac{4}{4a^2} > \frac{1}{4a^2}$$

Hence Option D

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Joined: 22 May 2016
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If 0 < a < 1, which of the following is the greatest? [#permalink]

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14 Sep 2017, 10:09
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Bunuel wrote:
If 0 < a < 1, which of the following is the greatest?

A. $$(-2a)^{(-2)}$$

B. $$\frac{1}{a^{(-2)}}$$

C. $$(\frac{1}{a})^{(-2)}$$

D. $$a^{(-2)}$$

E. $$a^2$$

Let a = $$\frac{1}{4}$$, and evaluate each answer choice. Straight algebra -- reciprocals, fractions, and squares -- became too tangled.

A. $$(-2a)^{(-2)}$$ --> $$\frac{1}{(-2*a)^2}$$

$$\frac{1}{(-2*\frac{1}{4})^2}$$ =

$$\frac{1}{(-\frac{1}{2})^2}$$ = $$\frac{1}{(\frac{1}{4})}$$ = 4

B. $$\frac{1}{a^{(-2)}}$$ --> $$\frac{a^2}{1^2}$$

$$a^2$$ = $$(\frac{1}{4})^2$$ =$$\frac{1}{16}$$

C.$$(\frac{1}{a})^{(-2)}$$ --> $$\frac{a^2}{1^2}$$=

$$a^2$$. Same as Answer B =$$\frac{1}{16}$$

D. $$a^{(-2)}$$--> $$\frac{1}{a^2}$$

$$\frac{1}{(\frac{1}{4})^2}$$ = $$\frac{1}{\frac{1}{16}}$$ = 16

E. $$a^2$$ = same as Answer B = $$\frac{1}{16}$$

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Re: If 0 < a < 1, which of the following is the greatest? [#permalink]

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20 Sep 2017, 14:51
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Expert's post
Bunuel wrote:
If 0 < a < 1, which of the following is the greatest?

A. $$(-2a)^{(-2)}$$

B. $$\frac{1}{a^{(-2)}}$$

C. $$(\frac{1}{a})^{(-2)}$$

D. $$a^{(-2)}$$

E. $$a^2$$

Since we know a is between 0 and 1, let’s let a = 1/2. Now we analyze each answer choice:

A)

[-2(1/2)]^-2 = (-1)^-2 = 1

B)

1/(1/2)^-2 = 1/(2^2) = ¼

C)

(1/(1/2))^-2 = 2^-2 = 1/4

D)

(1/2)^-2 = 2^2 = 4

E)

(1/2)^2 = 1/4

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Re: If 0 < a < 1, which of the following is the greatest?   [#permalink] 20 Sep 2017, 14:51
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