DS:-In-equalities
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- Brent@GMATPrepNow
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Target question: Is 0 < a < 1?
Given: a² = b
Statement 1: 0 < b < 1
There are several values of x and y that satisfy this condition. Here are two:
Case a: a = 0.5 and b = 0.25, in which case it IS true that 0 < a < 1
Case b: a = -0.5 and b = 0.25, in which case it is NOT true that 0 < a < 1
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: b < a
GIVEN: a² = b. From this, we can conclude that 0 < b
So, if b < a, then 0 < a (i.e., a is positive)
Let's consider two possible ranges for the value of a.
case a: 0 < a < 1 If this is the case, then a² < a
Since we're told that a² = b, then it must be true that b < a
So, case a yields no contradictions. That is, if 0 < a < 1, then a and b meet the given information as well as statement 2.
case b: 1 < a If this is the case, then a < a²
Since we're told that a² = b, then it must be true that a < b
Hmmm, this contradicts statement 2.
So, it's IMPOSSIBLE for 1 < a, and for a and b to meet the given information as well as statement 2.
So case a is the ONLY POSSIBLE CASE.
In other words, it must be true that 0 < a < 1
Since we can answer the target question with certainty, statement 2 is SUFFICIENT
Answer = B
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Brent
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If a²=b, is 1>a>0?
(1) 1>b>0
(2) a>b
Question stem rephrased: Is a a positive fraction?
An alternate approach is substitute a²=b into the statements.
Statement 1: 1>b>0
Substituting a²=b into 1>b>0, we get:
1>a²>0.
It's possible that a=1/2 or that a=-1/2.
INSUFFICIENT.
Statement 2: a>b
Substituting a²=b into a>b, we get:
a>a².
Implication: When a is squared, the result is a SMALLER value.
Only one type of number gets smaller when it's squared: a POSITIVE FRACTION.
SUFFICIENT.
The correct answer is B.
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If a^2 = b, then b is non-negative.
(1) If 1 > b > 0, then b is a proper fraction like ¼, whose square root would be both ±½. Hence insufficient.
(2) If a > b, and a^2 = b, then this happenstance is possible only when both a and b are positive proper fractions, hence 1 > a > 0 must receive a big [spoiler]YES. Sufficient, should learn about the literal meaning of a proper fraction.
Pick B[/spoiler]
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