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20 Mar 2022, 02:52
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If \(a > 0\), is \(t^a > w^a\) ?
(1) \(t > w\)
(2) \(t = 2w\)
Are You Up For the Challenge: 700 Level Questions
(1) \(t > w\)
(2) \(t = 2w\)
Are You Up For the Challenge: 700 Level Questions
28 Mar 2022, 02:35
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The given question is a "be verb" question. The answer to such questions will be yes or no.
Data is sufficient only if we can answer the question with a definite yes or a definite no.
From the question stem, we have a > 0 and we have to figure out whether \(t^a\) is greater than \(w^a\).
Because a > 0, \(t^a\) will be certainly be greater than \(w^a\) if t and w are positive and t > w.
Therefore, if we can determine two things from the statements, the data will be sufficient to answer the question
(i) We have to determine that both t and w are greater than 0 because if they are less than 0, the relation between \(t^a\) and \(w^a\) cannot be determined.
Why so?
For example if both t and w are negative and let us say t = -2 and w = -3.
Let us evaluate what happens when a = 3. \((-2)^3\) > \((-3)^3\)
However, when a = 2, \((-2)^2\) < \((-3)^2\)
(ii) We have to check whether t is greater than w.
Statement 1: t > w
This statement clearly states that t is greater than w. However, it does not gives us details about whether both t and w are positive.
Here are a couple of examples to highlight the existence of a counter argument.
Example: If t = 8, w = 4 and a = 2, then \(t^a\) > \(w^a\). We can answer the question with a definite yes.
Counter-Example: If t = -4 and w = -5 and a = 2, then \(t^a\) < \(w^a\). Answer to the question is No.
Because a counter example exists, we cannot find a definite answer to the question.
Statement 1 alone is not sufficient.
Eliminate answer options A and D.
Statement 2: t = 2w
Neither do we know whether t and w are positive nor do we know whether t > w
Here are a couple of examples to highlight the counter argument
Example: w = 3, t = 6 and a = 2; \(t^a\) > \(w^a\). Answer Yes.
Counter-Example: w = -3, t = -6 and a = 3; \(t^a\) < \(w^a\). Answer No.
Because a counter example exists, we cannot find a definite answer to the question.
Statement 2 alone is not sufficient.
Eliminate option B.
Combining Statement 1 and Statement 2 together.
We have t > w and t = 2w.
This means 2w > w implying that w > 0. Since t = 2w, then t > 0.
Thus we got answers to the two questions we were seeking answers to.
(i) t, w > 0 and (ii) t > w.
If t and w are positive and t > w, then definitely \(t^a\) > \(w^a\).
So Statements 1 and 2 together are sufficient to answer the question.
Choice C.
Data is sufficient only if we can answer the question with a definite yes or a definite no.
From the question stem, we have a > 0 and we have to figure out whether \(t^a\) is greater than \(w^a\).
Because a > 0, \(t^a\) will be certainly be greater than \(w^a\) if t and w are positive and t > w.
Therefore, if we can determine two things from the statements, the data will be sufficient to answer the question
(i) We have to determine that both t and w are greater than 0 because if they are less than 0, the relation between \(t^a\) and \(w^a\) cannot be determined.
Why so?
For example if both t and w are negative and let us say t = -2 and w = -3.
Let us evaluate what happens when a = 3. \((-2)^3\) > \((-3)^3\)
However, when a = 2, \((-2)^2\) < \((-3)^2\)
(ii) We have to check whether t is greater than w.
Statement 1: t > w
This statement clearly states that t is greater than w. However, it does not gives us details about whether both t and w are positive.
Here are a couple of examples to highlight the existence of a counter argument.
Example: If t = 8, w = 4 and a = 2, then \(t^a\) > \(w^a\). We can answer the question with a definite yes.
Counter-Example: If t = -4 and w = -5 and a = 2, then \(t^a\) < \(w^a\). Answer to the question is No.
Because a counter example exists, we cannot find a definite answer to the question.
Statement 1 alone is not sufficient.
Eliminate answer options A and D.
Statement 2: t = 2w
Neither do we know whether t and w are positive nor do we know whether t > w
Here are a couple of examples to highlight the counter argument
Example: w = 3, t = 6 and a = 2; \(t^a\) > \(w^a\). Answer Yes.
Counter-Example: w = -3, t = -6 and a = 3; \(t^a\) < \(w^a\). Answer No.
Because a counter example exists, we cannot find a definite answer to the question.
Statement 2 alone is not sufficient.
Eliminate option B.
Combining Statement 1 and Statement 2 together.
We have t > w and t = 2w.
This means 2w > w implying that w > 0. Since t = 2w, then t > 0.
Thus we got answers to the two questions we were seeking answers to.
(i) t, w > 0 and (ii) t > w.
If t and w are positive and t > w, then definitely \(t^a\) > \(w^a\).
So Statements 1 and 2 together are sufficient to answer the question.
Choice C.
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