Max@Math Revolution wrote:[Math Revolution GMAT math practice question]
The range of set A is 24 and the range of set B is 20. What is the smallest possible range of sets A and B, combined?
A. 20
B. 24
C. 40
D. 44
E. 48
$$? = \left( {{\rm{Rang}}{{\rm{e}}_{A \cup B}}} \right)\,\,\min $$
\[\left. \begin{gathered}
{\text{Rang}}{{\text{e}}_A} = 24{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \mathop \Rightarrow \limits^{{\text{WLOG}}{\mkern 1mu} \left( * \right)} \,\,\,\left\{ \begin{gathered}
\,\boxed{{x_1} - {x_2} = 24} \hfill \\
\,{x_j} - {x_k} \leqslant 24\,\,\,\,,\,\,\,{\text{for}}\,\,{\text{all}}\,\,\,{x_j},{x_k}\,\,{\text{in}}\,\,\,A \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,?\,\,\, \geqslant \,\,\,24 \hfill \\
\hfill \\
{\text{Rang}}{{\text{e}}_B} = 20{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \mathop \Rightarrow \limits^{{\text{WLOG}}{\mkern 1mu} \left( * \right)} \,\,\,\left\{ \begin{gathered}
\,{y_1} - {y_2} = 20 \hfill \\
\,{y_m} - {y_n} \leqslant 20\,\,\,\,,\,\,\,{\text{for}}\,\,{\text{all}}\,\,\,{y_m},{y_n}\,\,{\text{in}}\,\,\,B \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,?\,\,\, \geqslant \,\,\,20\,\,\, \hfill \\
\end{gathered} \right\}\,\,\,\,\,\,\boxed{\,\,{x_1}\,,\,\,{x_2}\,\,}\,\,\,\,\, \Rightarrow \,\,\,\,\,?\,\,\, \geqslant \,\,24\]
(*) WLOG = without loss of generality
\[{\text{Take}}\,\,\,\left\{ \begin{gathered}
\,{A_p} = \left\{ {24,0} \right\} \hfill \\
\,{B_p} = \left\{ {24,4} \right\} \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\left( {p = {\text{particular}}} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{A_p} \cup {B_p} = \left\{ {0,4,24} \right\}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\text{Rang}}{{\text{e}}_{{A_p} \cup {B_p}}}\, = 24\,\,\,\, \Rightarrow \,\,\,?\,\,\, \leqslant 24\,\]
\[\left\{ \begin{gathered}
\,\,?\,\,\, \geqslant \,\,\,24 \hfill \\
\,\,?\,\,\, \leqslant \,\,\,24 \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,? = 24\]
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
