swerve wrote:A tank has 5 inlet pipes. Three pipes are narrow and two are wide. Each of the three narrow pipes works at 1/2 the rate of each of the wide pipes. All the pipes working together will take what fraction of time taken by the two wide pipes working together to fill the tank?
A. 1/2
B. 2/3
C. 3/4
D. 3/7
E. 4/7
Source: Veritas Prep
\[5\,\,{\text{pipes}}\,\,\,\left\{ \begin{gathered}
\,3\,\,{\text{narrow}}\,\,\,\, \to \,\,\,{\text{each}}\,\,\,1\,\,{\text{gallons}}/\min \,\,\, \hfill \\
\,2\,\,{\text{wide}}\,\,\,\,\,\,\,\,\, \to \,\,\,{\text{each}}\,\,\,2\,\,{\text{gallons}}/\min \hfill \\
\end{gathered} \right.\,\,\,\,\,\left( {{\text{particular}}\,\,{\text{case}}!} \right)\]
\[{\text{A}}\,\,\,{\text{ = }}\,\,\,{\text{2}}\,\,{\text{wide}}\,\,{\text{together}}\,\,\,{\text{:}}\,\,\,\,2 \cdot 2 = 4\,\,{\text{gallons/min}}\]
\[{\text{B}}\,\,\,{\text{ = }}\,\,\,{\text{all}}\,\,{\text{5}}\,\,{\text{together}}\,\,\,{\text{:}}\,\,\,\,3 \cdot 1 + 2 \cdot 2 = 7\,\,{\text{gallons/min}}\]
\[{\text{B:A}}\,\,\underline {{\text{work}}} \,\,{\text{ratio}}\,\,\left( {{\text{per}}\,\,{\text{any}}\,\,{\text{time}}} \right)\,\,\,{\text{ = }}\,\,\,\,\frac{7}{4}\,\,\,\,\,\]
\[?\,\,\, = \,\,\,B:A\,\,\underline {{\text{time}}} \,\,{\text{ratio}}\,\,\,\left( {{\text{per}}\,\,{\text{any}}\,\,{\text{work}}} \right)\,\,\, = \,\,\,{\left( {\frac{7}{4}} \right)^{ - 1}} = \,\,\,\frac{4}{7}\]
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
