If n is an integer, is (0.1)^n greater than (10)^n?

Target question: Is (0.1)^n greater than (10)^n?

REPHRASED target question: Is (1/10)^n greater than (10)^n?

Statement 1: n > −10
This statement doesn't FEEL sufficient, so I'll TEST some values.
There are several values of n that satisfy statement 1. Here are two:
Case a: n = 1, in which case (1/10)^n is NOT greater than (10)^n
Case b: n = -1, in which case (1/10)^-1 = 10 and 10^-1 = 1/10. Here, (1/10)^n IS greater than (10)^n
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Aside: For more on this idea of plugging in values when a statement doesn't feel sufficient, you can read my article: https://www.gmatprepnow.com/articles/dat ... lug-values

Statement 2: n < 10
There are several values of n that satisfy statement 2. Here are two:
Case a: n = 1, in which case (1/10)^n is NOT greater than (10)^n
Case b: n = -1, in which case (1/10)^-1 = 10 and 10^-1 = 1/10. Here, (1/10)^n IS greater than (10)^n
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

IMPORTANT - Notice that I tested the SAME VALUES for both statements. This means that, the STATEMENTS COMBINED are also NOT SUFFICIENT

Answer = E

Cheers,
Brent

If n is an integer, is (0.1)^n greater than (10)^n?

(1) n > −10
(2) n < 10

Sol. (10)^-n > (10)^n or -n>n or n<0 ?
1) n > -10 Not sufficient
2) n < 10 Not sufficient

1) + 2) -10 < n <10 Not sufficient

Hence E

Please note that for n=0, both the expressions will be equal to 1.

Hence, Answer: E.

Here's another approach:

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?

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:
RELATED VIDEOS

Bunuel can we play around with the statement in question like this?
I mean to say:

Its given that is 10^-n>10^n

So can we divide/subtract/multiple/add on both sides of an expression that is to be proved?

When should we not do it?


Thanks

How to manipulate inequalities (adding, subtracting, squaring etc.).

Video solution from Quant Reasoning:
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Solution:

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

Answer: Option E

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IMO here is the simplest explanation:

Let’s analyze the question --------- the only case when 0.1^n > 10^n is if n is negative
So basically, we are trying to find if n < 0

Statement 1:
N > -10
Here n can be ------- n < 0 or if the value goes beyond 0 then n > 0
NOT SUFFICIENT

Statement 2:
N < 10
Here n can be --------- n > 0 or n < 0
NOT SUFFICIENT

Statement 1+2
-10 < n < 10
Again n can be ------- n < 0 or n > 0
NOT SUFFICIENT

Answer – E

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