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If a and b are positive numbers, is a < b ?
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26 Jan 2022, 04:00
26 Jan 2022, 04:00
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If a and b are positive numbers, is \(a < b\)?
(1) \(a < \frac{b}{2} + 2\)
(2) \(a < \frac{b}{2} -2\)
Are You Up For the Challenge: 700 Level Questions
(1) \(a < \frac{b}{2} + 2\)
(2) \(a < \frac{b}{2} -2\)
Are You Up For the Challenge: 700 Level Questions
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Re: If a and b are positive numbers, is a < b ?
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27 Jan 2022, 09:21
27 Jan 2022, 09:21
Expert Reply
Top Contributor
Bunuel wrote:
If a and b are positive numbers, is \(a < b\)?
(1) \(a < \frac{b}{2} + 2\)
(2) \(a < \frac{b}{2} -2\)
Are You Up For the Challenge: 700 Level Questions
(1) \(a < \frac{b}{2} + 2\)
(2) \(a < \frac{b}{2} -2\)
Are You Up For the Challenge: 700 Level Questions
Given: a and b are positive numbers
Target question: Is \(a < b\) ?
Statement 1: \(a < \frac{b}{2} + 2\)
Multiply both sides of the inequality by \(2\) to get: \(2a < b + 4\)
Subtract \(4\) from both sides: \(2a - 4 < b\)
There are several values of a and b that satisfy the inequality \(2a - 4 < b\). Here are two:
Case a: \(a = 1\) and \(b = 10\). In this case, the answer to the target question is YES, a < b
Case b: \(a = 2\) and \(b = 1\). In this case, the answer to the target question is NO, a is not less than b
Since we can’t answer the target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: \(a < \frac{b}{2} -2\)
Multiply both sides of the inequality by \(2\) to get: \(2a < b - 4\)
Add \(4\) to both sides: \(2a + 4 < b\)
Since \(2a < 2a + 4\), we can add this to our inequality to get: \(2a < 2a + 4 < b\)
Useful property: If \(k > 0\), then \(k < 2k \)
Since we're told \(a\) is positive, we can add another component to our inequality to get: \(a < 2a < 2a + 4 < b\)
At this point, it's clear that a < b, which means the answer to the target question is YES, a < b
Since we can answer the target question with certainty, statement 2 is SUFFICIENT
Answer: B
Cheers,
Brent
gmatclubot
Re: If a and b are positive numbers, is a < b ? [#permalink]
27 Jan 2022, 09:21
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