Problem Solving

[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

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.

=>

Note that the range of the combined set can't be less than the maximum of the ranges of the two sets.

The maximum of the ranges of the two sets is 24. So, the range of the combined set must be greater than or equal to 24.
For example, if A = {0, 24} and B = { 0, 20 }, the combined set { 0, 20, 24 }, has range 24.
24 is the smallest possible range of the set A ⋃ B.

Therefore, B is the answer.
Answer: B