Question

If a, b, c, and d are positive numbers, is $\frac{a}{b}<\frac{c}{d}$?
(1) $0<\frac{c-a}{d-b}$
(2) $(\frac{ad}{dc}{)}^2<\frac{ad}{bc}$

• A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Each statement alone is sufficient.
• E. Statements (1) and (2) together are not sufficient.

Answer 1

Answer: (B)

Statement (2) alone is sufficient, but statement (1) alone is not sufficient.

Determine whether , where a, b, c, and d are positive numbers.

1. Given that , then a = 2, b = 3, c = 6, and d = 8 are possible values for a, b, c, and d because and . For these values, is true because . On the other hand, a = 4, b = 6, c = 2, and d = 3 are also possible values of a, b, c, and d because and . For these values, is false because ; NOT sufficient.
2. Given that , then
   

• , dividing both sides by the positive number .
  
• , multiplying both sides by the positive number c.
  
• , dividing both sides by the positive number d; SUFFICIENT.
  
• The correct answer is B; statement 2 alone is sufficient.