Is a < b? (1) a^b < b^a (2) a/b > 1

1. a^b < b^a

Let's structure possible variants:
1.1 a<b, let's prove that this variant exists
a = 1
b = 2
than a^b < b^a hold true (1<2)
1.2 a>b, let's prove that this variant exists
a=4
b=3
than a^b < b^a hold true (64<81)
So, this condition is INSUFFICIENT

2. a/b > 1

Let's structure possible variants:
2.1 a>0, b > 0 => a>b
2.2 a<0, b < 0 => a<b
INSUFFICIENT as well as 1 condition

1&2
So, variant than a>b exists because of logic in point 1.2 and 2.1
BUT we don't know about a<b, because in point 1.1 we use positive integer, but in point
2.2 is a condition that a < 0 and b < 0
So let's check does this variant exist?
a = -2
b = -1
Try 1 condition (a^b < b^a):
(-2)^(-1) < (-1)^(-2)
-1/2 < 1
OK, we prove it and that mean we don't khow a>b or a<b.

Answ E

This is a DS question where, although the question stem is quite straightforward in what it asks, the statements are tricky to interpret.
The best approach, therefore, is to plug in simple values and check.

From statement I alone, we know \(a^b < b^a\).

If a = 2 and b = 3, \(2^3 < 3^2\) is true. Also, a<b. So, we get a YES as an answer to the main question.

If a = -2 and b = -3, \((-2)^{-3} < (-3)^{-2}\) i.e. (-\(\frac{1}{8}\)) < (\(\frac{1}{9}\)) is true. But here, a>b and we get a NO as an answer to the main question.

Statement I alone is insufficient. Answer options A and D can be eliminated. Possible answers are B, C or E.

Statement II alone says a/b > 1. A common mistake that some students do when they see this inequality is that they cross multiply b with 1 and conclude a>b. This is wrong because we do not know the signs of a and b.

a/b > 1 can mean two things:

Both a and b are positive and a>b.

Both a and b are negative and hence a<b.

So, this is insufficient to answer the main question with a definite YES or NO. Option B can be eliminated.

When we combine both the statements, we can use the following values for a and b:
If a = 5 and b = 4 (taking slightly bigger values to accommodate the fact that \(a^b < b^a\)), \(5^4 < 4^5\) i.e. 625 < 1024 is true. Here a>b.
If a = -2 and b= -1, \((-2)^{-1}\) = -\(\frac{1}{2}\) and \((-1)^{-2}\) = 1. Naturally, \(a^b\) (negative) < \(b^a\) (positive). But, here a<b.

Even after combining the statements, we don’t have a unique YES or NO to the question. So,the correct answer is E.

Hope this helps!

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