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If n is an integer, is (0.1)^n greater than

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jjjinapinch Senior | Next Rank: 100 Posts Default Avatar
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If n is an integer, is (0.1)^n greater than

Post Wed Aug 02, 2017 11:04 am
If n is an integer, is (0.1)^n greater than (10)^n?
(1) n > -10
(2) n < 10

Official Guide question
Answer: E

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Post Wed Aug 02, 2017 12:01 pm
jjjinapinch wrote:
If n is an integer, is (0.1)^n greater than (10)^n?
(1) n > -10
(2) n < 10

Official Guide question
Answer: E
Target question: Is (0.1)^n > (10)^n?
This is a good candidate for rephrasing the target question.
Since (0.1)^n is always POSITIVE, we can safely divide both sides of the inequality by (0.1)^n to get: 1 > [(10)^n]/[(0.1)^n]
There's a nice rule that says (a^n)/(b^n) = (a/b)^n
When we apply this rule to the right side of the inequality, we get: 1 > (10/0.1)^n
Simplify to get: Is 1 > 100^n?
Notice that, when n = 0, then 100^n = 100^0 = 1
So, when n > 0, then 100^n > 1, and when n < 0, then 100^n < 1
So, we can REPHRASE the target question as....
REPHRASED target question: Is n < 0?

Aside: Here’s a video with tips on rephrasing the target question: http://www.gmatprepnow.com/module/gmat-data-sufficiency?id=1100

Statement 1: n > -10
There are several values of n that satisfy statement 1. Here are two:
Case a: n = -9, in which case n < 0
Case b: n = 2, in which case n > 0
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: n < 10
There are several values of n that satisfy statement 1. Here are two:
Case a: n = -9, in which case n < 0
Case b: n = 2, in which case n > 0
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
IMPORTANT: Notice that I was able to use the same counter-examples to show that each statement ALONE is not sufficient. So, the same counter-examples will satisfy the two statements COMBINED.
Since we cannot answer the REPHRASED target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

Cheers,
Brent

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Top Reply
Post Wed Aug 02, 2017 11:57 am
When you don't have a calculator, it's often easier to convert decimals into fractions, particularly when dealing with exponents:
Is (1/10)^n greater than (10)^n?

Ask yourself: when would a fraction raised to a power be greater than an integer raised to the same power? If the exponent were negative! A negative exponent is equal to the reciprocal of the positive exponent:
(1/10)^(-1) = 10
10^(-1) = 1/10

If n > 0, then (1/10)^n < 10^n
If n = 0, then (1/10)^n = 10^n
If n < 0, then (1/10)^n > 10^n

Target question: is n < 0 ?

(1) n > -10
n could be negative, if n = -9, or it could be any integer greater than or equal to 0. Insufficient.

(2) n < 10
Again, n could be negative (this range includes all negatives), or any positive integer less than 10. Insufficient.

(1) & (2) together:
-10 < n < 10
This range includes both negatives and non-negatives, so we do not get a definitive answer to our question. Insufficient.

The answer is E.

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