If n is an integer, is (0.1)^n greater than (10)^n?
(1) n > -10
(2) n < 10
Official Guide question
Answer: E
If n is an integer, is (0.1)^n greater than
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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.
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.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
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Target question: Is (0.1)^n > (10)^n?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
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: https://www.gmatprepnow.com/module/gmat- ... cy?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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Solution:jjjinapinch wrote: ↑Wed Aug 02, 2017 11:04 amIf n is an integer, is (0.1)^n greater than (10)^n?
(1) n > -10
(2) n < 10
Official Guide question
Answer: E
Question Stem Analysis:
We need to determine whether 0.1^n is greater than 10^n, given that n is an integer. Notice that 0.1 = 10^(-1); therefore, 0.1^n = 10^(-n) and 10^(-n) is greater than 10^n if -n > n. We see that -n > n if and only if n is negative. In other words, we need to determine whether n is negative.
Statement One Alone:
Even though we know n > -10, n could be either positive or negative. So we cannot definitely say n is negative. Statement alone is not sufficient.
Statement Two Alone:
Even though we know n < 10, n could be either positive or negative. So we cannot definitely say n is negative. Statement two is not sufficient.
Statements One and Two Together:
With the two statements, we see that -10 < n < 10. Thus, n could still be either positive or negative. Both statements together are not sufficient.
Answer: E
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