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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 < 0 We 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 < 0 From 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
b < x < 0 --> b and x are negative. b is greater in magnitude than x.
w < x < y --> w and x are negative. w is greater in magnitude than x. We do not know anything about y.
I. (w + b)/y < 0 --> -ve/(Don't know whether y is positive or negative). Not a must be true statement.
II. (y – b)/b < 0 --> If y is negative then b is greater in magnitude than y and y - b will be positive. If y is positive then y - b will always be positive. In both the cases (y - b)/b = +ve/-ve = -ve. Hence its a must be true statement.
III. (b + w) – (x + y) < 0 --> Magnitude wise b + w > x --> b + w - x = -ve We have to find the sign of -ve - y If y is negative the b + w - x is greater in magnitude than y --> The expression is negative. If y is positive then the overall expression is negative. Hence its a must be true statement.
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
*Kudos for all correct solutions
(1) w+b<0 but y can be >0 or <0 not must be true.
(2) y-b = if y>0 then y-b>0 or if y<0 then y-b>0 and b<0 thus y-b/b<0 must be true
(3)b +w has more magnitude than x+y if y<0 Also if y>0 and more magnitude than b+w then entire equation is more negative value.. thus it must be true for every case suff
b < x < 0 and w < x < y With the above info, we are pretty sure that b,x,w <0 . We dont know about y , whether it is >0, 0, or <0. 1. (w+b)/y <0 we know that w+b is <0 , but given equation varies with y. So it must not be true 2. (y-b)/b <0, we know y>b & b<0.= > y-b >0 & b<0 . So the given equation always holds & must be true. 3. (b+w) - (x+y)<0 the given equation can be written as (b-x) - (w-y)<0 we know, from given data in the question, that b-x<0 & w-y<0 therefore, subtraction of two negative numbers always be negative. So, Given equation must be true.