# Given $a=bc$ and $c\geq 1$ and $b\neq 0$, which is greater: $a$ or $(b+c)$?

I am trying a GRE question, and I have the following:

Given $a=bc$ and $c\geq 1$ and $b\neq 0$,

Answer A: $a$ > $(b+c)$

Answer B: $a$ < $(b+c)$

Answer C: $a$ = $(b+c)$

Answer D: The relationship cannot be determined from the info given.

Now, trying to reason out the problem statement, we get:

$a \geq b$ and $b+c=b+\frac{a}{b}$ which gives $b+c=b$+{some value greater than or equal to 1}

Now, my hunch is that the answer is option D, but for whatever values I try to plug in, I seem to get option B. Is there a way to say with certainty that option B is the correct answer?

• What about the sign of $b$? – mrs Jan 22 '14 at 11:30
• $b$ is nonzero. – Joebevo Jan 22 '14 at 11:31

• If $b=2, c=4$ then $a=8$, so the first one would be right.
• If $b=-2, c=4$ then $a<b+c$ so the second one could be right. So I think D is our choice.
Take: $$b = -0.5$$ $$c = 1$$ Then : $$a = -0.5$$ In this case , $a = b + c = -0.5$ Similarly prove that for some example options $A,B$ also hold true and hence option $D$ is correct.