If \(d > 0\) and \(0 < 1 - \dfrac{c}{d} < 1,\) which of the following must be true?

Gmat_mission wrote: ↑

Thu Apr 29, 2021 7:34 am

If \(d > 0\) and \(0 < 1 - \dfrac{c}{d} < 1,\) which of the following must be true?

I. \(c > 0\)
II. \(\dfrac{c}{d} < 1\)
III. \(c^2 + d^2 > 1\)

(A) I only
(B) II only
(C) I and II only
(D) II and III only
(E) I, II, and III

Answer: C

Source: Official Guide

Solution:

Let’s simplify the double inequality:

0 < 1 - c/d < 1

-1 < -c/d < 0

Multiplying both sides by -d (which is a negative quantity since d > 0), we have:

d > c > 0

We see that c > 0, so statement I is true. Now, dividing both sides by d, we have:

1 > c/d > 0

We see that c/d < 1, so statement II is true.

However, statement III might not be true. For example, if d = 0.4 and c = 0.3, then c^2 + d^2 = 0.09 + 0.16 = 0.25, which is not greater than 1.

Answer: C