BTGmoderatorDC wrote:If x and y are positive integers, is x/y < (x+5)/(y+5)?
(1) y = 5
(2) x > y
Source: Magoosh
$$x,y\,\, \ge 1\,\,\,{\rm{ints}}$$
$${x \over y}\,\,\mathop < \limits^? \,\,{{x + 5} \over {y + 5}}$$
$$\left( 1 \right)\,\,y = 5\,\,\,\,\, \Rightarrow \,\,\,\,\,{x \over 5}\,\,\mathop < \limits^? \,\,{{x + 5} \over {10}}\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{ \cdot \,\,10} \,\,\,\,\,\,\,\,2x\,\,\mathop < \limits^? \,\,x + 5$$
$$\left\{ \matrix{
\,{\rm{Take}}\,\,x = 1\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,\,\,\,\,\,\,\,\,\, \hfill \cr
\,{\rm{Take}}\,\,x = 5\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\, \hfill \cr} \right.$$
$$\left( 2 \right)\,\,\,{x \over y}\,\,\mathop < \limits^? \,\,{{x + 5} \over {y + 5}}\,\,\,\,\,\,\,\,\,\mathop \Leftrightarrow \limits^{y\,\,\left( {y + 5} \right)\,\, > \,\,0} \,\,\,\,\,\,\,x\,\left( {y + 5} \right)\,\,\mathop < \limits^? \,\,y\left( {x + 5} \right)\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x\,\mathop < \limits^? \,y\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,$$
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
