If 0 < a < b, is c < 0? (1) cb < ca (2) a – b > c

Target question: Is c < 0?

Given: 0 < a < b
If a < b, then we know that b - a > 0
In other words, b - a = some positive number
It also means that a - b = some negative number

Statement 1: cb < ca
Subtract ca from both sides to get: cb - ca < 0
Factor: c(b - a) < 0
So, c(some positive number) < 0
This means that c must be a negative number.
In other words, c < 0
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: a – b > c
So, some negative number > c
If c is less than some negative number, then c must be a negative number.
In other words, c < 0
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer:

ASIDE: For statement 1, once we get to c(some positive number) < 0, we can also just divide both sides by some positive number to get c < 0

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Is c <0?

Statement 1: c(b-a)<0
But since b>a>0 so b-a will be positive
Hence c must be negative
Sufficient

Statement 2: a-b>c
a-b is negative as a<b
I.e. c<negative
Hence c is definitely negative
Sufficient

Answer Option D
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We are given that 0 < a < b and we need to determine whether c is negative.

Statement One Alone:

cb < ca

Since we know that a and b are positive and that b is greater than a, we see that c must be negative. For example, if a = 2 and b = 3, in order for cb < ca, c could be -1, such that -3 < -2. Statement one alone is sufficient to answer the question.

Statement Two Alone:

a – b > c

Since a is less than b and they are both positive, (a - b) must be negative; thus, we have:

Negative number > c

Since c is less than some negative number, c is also negative. Statement two is sufficient to answer the question.

Answer: D
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0 < a < b implies the following:
b-a = bigger - smaller = POSITIVE
a-b = smaller - bigger = NEGATIVE

Statement 1:
cb - ca < 0
c(b-a) < 0
Implication:
c and b-a have DIFFERENT SIGNS.
Since b-a is positive, c must be negative.
SUFFICIENT.

Statement 2:
c < a-b
c < negative
Thus, c must be negative.
SUFFICIENT.

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