Re: If a, b, c and d are positive integers and a/b < c/d, which

If a, b, c and d are positive integers and a/b < c/d, which of the following must be true?

I. (a+c)/(b+d) < c/d
II. (a+c)/(b+d) < a/b
III. (a+c)/(b+d) = a/b + c/d

A. None
B. I only
C. II only
D. I and II
E. I and III

Since the numbers are positive we can safely cross-multiply. So, we are given that ad<bc.

I. \frac{a+c}{b+d} < \frac{c}{d} --> ad+cd<bc+cd--> cd cancels out: ad<bc. This is given to be true.

II. \frac{a+c}{b+d} < \frac{a}{b} --> ab+bc<ab+ad --> ab cancels out: bc<ad. Opposite what is given, thus this option is not true.

III. \frac{a+c}{b+d} = \frac{a}{b} + \frac{c}{d}. From I we already know that \frac{a+c}{b+d} < \frac{c}{d}, thus when we add a positive value (\frac{a}{b}) to \frac{c}{d} we make the right hand side even bigger. Thus \frac{a+c}{b+d} = \frac{a}{b} + \frac{c}{d} cannot be true.

Answer: B.