GMATPrepNow
If b < x < 0 and w < x < y, then which of the following MUST be true?
I. \(\frac{w + b}{y} < 0\)
II. \(\frac{y – b}{b} < 0\)
III. \((b + w) – (x + y) < 0\)
(A) II only
(B) III only
(C) I and II only
(D) I and III only
(E) II and III only
Here's a different approach:
I. (w + b)/y < 0We know that w and b are NEGATIVE, so (w + b) = NEGATIVE
However, we don't know whether y is NEGATIVE or POSITIVE
As such, (w + b)/y can be either POSITIVE or NEGATIVE
So, statement 1 need not be true.
II. (y – b)/b < 0From the given information, we know that b < y. So, if we subtract b from both sides, we get y - b > 0
In other words, y-b is POSITIVE
Since we also know that b is NEGATIVE, we can see that (y – b)/b = POSITIVE/NEGATIVE = NEGATIVE
So, statement II must be true
III. (b + w) – (x + y) < 0 First rewrite the inequality as: b + w - x - y < 0
The rewrite as: (b - x) + (w - y) < 0
So, statement III can be rewritten as: (b - x) + (w - y) < 0
From the given information, we know that b < x.
If we subtract x from both sides, we get: b - x < 0
In other words, b-x is NEGATIVE
Also, from the given information, we know that w < y.
If we subtract y from both sides, we get: w - y < 0
In other words, w-y is NEGATIVE
This means that (b - x) + (w - y) = NEGATIVE + NEGATIVE = NEGATIVE
So, it must be true that (b - x) + (w - y) < 0
Answer: E
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