If a and b are negative, and c is positive, which of the following statements is/are true?
I. a - b < a - c
II. if $a<b,then\frac{a}{c}<\frac{b}{c}$
III. $\frac{1}{b}<\frac{1}{c}$

The correct option is D II and III only
Best way to solve this problem is by simply assuming some values for a, b, and c.
Let a = -5, b = -2, and c = 3
Now check for all the three statements.
(I) a - b < a - c
a - b = -5 - (-2) = -3 and a - c = -5 - 3 = -8
Clearly -3 < -8, so first statement is wrong.
Similarly, check for statement (II) and (III).
we find that statement (II) and (III) are true.
$\therefore$ Correct answer is option (d).

Alternate Approach:
Let a = - x, b = - y and c = z
Statement I: - x + y < - x - z
So I is not true.
Statement II : $\frac{-x}{z}<\frac{-y}{z}$
Since x < y so II is true
Statement III : $\frac{1}{-y}<\frac{1}{z}$
Since y is negative and z is positive.
So III is true
Hence, statements II and III are true.