BTGmoderatorDC wrote:If a and b are positive integers, is a/b = 5/8 ?
(1) 1/2 < a/b < 2/3
(2) b = 8
Source: Princeton Review
$$a,b\,\, \ge 1\,\,{\rm{ints}}$$
$${a \over b}\,\,\mathop = \limits^? \,\,{5 \over 8}$$
$$\left( 1 \right)\,\,{1 \over 2} < {a \over b} < {2 \over 3}\,\,\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {5,8} \right)\,\,\,\,\,\left( * \right) \hfill \cr
\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {50001,80000} \right)\,\,\,\,\,\left( {**} \right) \hfill \cr} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,{\rm{INSUFF}}.$$
$$\left( * \right)\,\,{1 \over 2} < {5 \over 8}\,\,\,\left( { = {{4 + 1} \over 8} = {1 \over 2} + {1 \over 8}} \right)\,\,\,\,\,{\rm{AND}}\,\,\,\,\,\,{5 \over 8}\,\, < {2 \over 3}\,\,\,\,\,\left( {5 \cdot 3 < 8 \cdot 2} \right)\,\,\,$$
$$\left( {**} \right)\,\,\,{\rm{Nice}}\,\,{\rm{idea}}:\,\,\,{{50001} \over {80000}}\,\,\,{\rm{is}}\,{\rm{not}}\,\,\,{5 \over 8}\,\,\,{\rm{but}}\,\,\,{\rm{VERY}}\,\,\,{\rm{NEAR}}\,\,\,{\rm{it}}\,!$$
$$\left( 2 \right)\,\,b = 8\,\,\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {5,8} \right)\,\,\,\,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {1,8} \right)\,\,\,\,\, \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,{\rm{INSUFF}}.$$
$$\left( {1 + 2} \right)\,\,\,\,\boxed{b = 8}\,\,\,;\,\,\,\,\,\frac{1}{2} < \frac{a}{8} < \frac{2}{3}\,\,\,\,\,\,\mathop \Rightarrow \limits^{ \cdot \,\,\left( {8 \cdot 3} \right)} \,\,\,\,12 < 3a < 16\,\,\,\,\mathop \Rightarrow \limits^{a\,\,\operatorname{int} } \,\,\,\,\boxed{a = 5}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,{\text{SUFF}}.$$
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
