BTGmoderatorDC wrote:For integers a, b, c, a/(b - c)=1 what is the value of (b-c)/b ?
(1) a/b=3/5
(2) a and b have no common factors greater than 1
Source: Veritas Prep
\[\left\{ \begin{gathered}
\,a,b,c\,\,{\text{ints}}\,\, \hfill \\
\,\frac{a}{{b - c}} = 1 \hfill \\
\end{gathered} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,a = b - c\,\,\,\,\,\left( * \right)\]
\[? = \frac{{b - c}}{b}\]
\[\left( 1 \right)\,\,\frac{a}{b} = \frac{3}{5}\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\left( {? = } \right)\,\,\frac{{b - c}}{b} = \frac{3}{5}\,\,\,\]
\[\left( 2 \right)\,\,\,GCF\left( {a,b} \right) = 1\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,\left( {a,b,c} \right) = \left( {1,2,1} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,{\text{?}}\,\, = \frac{1}{2} \hfill \\
\,{\text{Take}}\,\,\left( {a,b,c} \right) = \left( {1,3,2} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,{\text{?}}\,\, = \frac{1}{3}\,\, \hfill \\
\end{gathered} \right.\]
Important: to offer a viable BIFURCATION, we must also control the value of c, so that all restrictions are shown to be obeyed.
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
