nkmungila wrote:

If \(\frac{a}{bc}\) is positive, which of the following must be true?

A. a + b + c > 0

B. bc > 0

C. a > 0 and bc > 0

D. a < 0 and bc < 0

E. abc > 0

With algebra, I eliminated answers, and chose E. But the manipulation of abstractions can make me wary.

So I chose numbers (mindful of Answer A, I kept absolute values close together)

Testing numbers was much easier.

Four iterations yield +2

1) \(\frac{4}{(2)(1)}\)

2) \(\frac{4}{(-2)(-1)}\)

3) \(\frac{-4}{(-2)(1)}\)

4) \(\frac{-4}{(2)(-1)}\)

C (per #s 1 and 2) and D (per #s 3 and 4) can be true. But each proves that the other is not a case of "must be true."

A is contradicted by case #4:

-4 + (-1) + 2 = -3

B is contradicted by #3 and #4:

(-2)(1) = -2 and (2)(-1) = -2

Remaining: E) abc > 0

All four cases: true.

Answer E

_________________

In the depths of winter, I finally learned

that within me there lay an invincible summer.

-- Albert Camus, "Return to Tipasa"