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100 points for $49 worth of Veritas practice GMATs FREE VERITAS PRACTICE GMAT EXAMS Earn 10 Points Per Post Earn 10 Points Per Thanks Earn 10 Points Per Upvote ## Tim and Robert have entered a race, the rules of which tagged by: AAPL ##### This topic has 3 expert replies and 0 member replies ### Top Member ## Tim and Robert have entered a race, the rules of which ## Timer 00:00 ## Your Answer A B C D E ## Global Stats Difficult Veritas Prep Tim and Robert have entered a race, the rules of which stipulate that each runner must run for at least 4 hours and no runner can run for more than 6 hours. Together, they must run a total of 50 miles. If it takes Tim 15 minutes to run a mile and Robert 12 minutes to run a mile, what is the minimum number of miles Robert must run if both Tim and Robert must individually run a whole number of miles? A. 18 B. 20 C. 22 D. 24 E. 26 OA E. ### GMAT/MBA Expert GMAT Instructor Joined 09 Oct 2010 Posted: 1449 messages Followed by: 32 members Upvotes: 59 AAPL wrote: Veritas Prep Tim and Robert have entered a race, the rules of which stipulate that each runner must run for at least 4 hours and no runner can run for more than 6 hours. Together, they must run a total of 50 miles. If it takes Tim 15 minutes to run a mile and Robert 12 minutes to run a mile, what is the minimum number of miles Robert must run if both Tim and Robert must individually run a whole number of miles? A. 18 B. 20 C. 22 D. 24 E. 26 Let T and R be the number of minutes Tim and Robert run, respectively. Hence: DATA: $$4 \cdot 60\,\,\, \leqslant \,\,\,\,\,T,R\,\,\,\, \leqslant \,\,\,6 \cdot 60\,\,\,\,\,\,\,\left( {\text{I}} \right)$$ $\left. \begin{gathered} \operatorname{int} \,\,\, = \,\,\,T\,\,\min \,\,\,\left( {\frac{{1\,\,{\text{mile}}}}{{15\,\,\min }}} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,T\,\,{\text{divisible}}\,\,{\text{by}}\,\,15\,\,\,\,\, \hfill \\ \operatorname{int} \,\,\,\mathop = \limits^{\left( * \right)} \,\,\,R\,\,\min \,\,\,\left( {\frac{{1\,\,{\text{mile}}}}{{12\,\,\min }}} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,R\,\,{\text{divisible}}\,\,{\text{by}}\,\,12 \hfill \\ \end{gathered} \right\}\,\,\,\,\,\,\,\left( {{\text{II}}} \right)$ $\boxed4\,T\,\,\min \,\,\,\left( {\frac{{1\,\,{\text{mile}}}}{{3 \cdot 5 \cdot \boxed4\,\,\min }}} \right)\,\,\, + \,\,\boxed5\,R\,\,\min \,\,\,\left( {\frac{{1\,\,{\text{mile}}}}{{3 \cdot 4 \cdot \boxed5\,\,\min }}} \right) = \frac{{50 \cdot \boxed{3 \cdot 4 \cdot 5}}}{{\boxed{3 \cdot 4 \cdot 5}}}\,\,{\text{miles}}$ $4T + 5R = 50 \cdot 3 \cdot 4 \cdot 5\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,4T = 5\left( {3 \cdot 4 \cdot 5 \cdot 10 - R} \right)\,\,\,\,\mathop \leqslant \limits^{\left( {\text{I}} \right)} \,\,\,\,4 \cdot 6 \cdot 60$ $R\,\,\, \geqslant \,\,\,3 \cdot 4 \cdot 5 \cdot 10 - 4 \cdot 6 \cdot 12 = 4 \cdot 6 \cdot \left( {25 - 12} \right) = 4 \cdot 6 \cdot 13\,\,\,\,\,\,\,\left( {{\text{III}}} \right)$ FOCUS: $?\,\,\,:\,\,\,\min \,\,\,\frac{R}{{12}}\,\,\,\,\,\left( * \right)\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\min \,\,\,R\,\,\,\,\,\,\,\,\,\,\,\,\left[ {\,? = {{\left( {\frac{R}{{12}}} \right)}_{\,\min }}\,} \right]\,\,\,\,\,$ DATA-FOCUS CONNECTION: $$\left. \matrix{ \left( {\rm{I}} \right)\,\,\, \Rightarrow \,\,\,R \ge 4 \cdot 60 = 4 \cdot 3 \cdot 20\,\,\, \hfill \cr \left( {{\rm{II}}} \right)\,\,\, \Rightarrow \,\,\,R = 3 \cdot 4 \cdot {\mathop{\rm int}} \hfill \cr \left( {{\rm{III}}} \right)\,\,\, \Rightarrow \,\,\,R \ge 2 \cdot 3 \cdot 4 \cdot 13 \hfill \cr} \right\}\,\,\,\,\, \Rightarrow \,\,\,\,\,? = {\mathop{\rm int}} = 26$$ $\left( \begin{gathered} {R_{\,\min }} = \underline {3 \cdot 4 \cdot 26} \,\,\,\, \Rightarrow \,\,\,4T = 5\left( {3 \cdot 4 \cdot 5 \cdot 10 - \underline {3 \cdot 4 \cdot 26} } \right) = \underleftrightarrow {60\left( {50 - 26} \right)} = 60 \cdot 24 \hfill \\ T = 6 \cdot 60\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\left\{ \begin{gathered} \left( {\text{I}} \right)\,\,\,{\text{ok}} \hfill \\ \left( {{\text{II}}} \right)\,\,\,{\text{ok}} \hfill \\ \end{gathered} \right.\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,{R_{\,\min }} = 3 \cdot 4 \cdot 26\,\,\,{\text{viable}}\,\,\,\, \hfill \\ \end{gathered} \right)$ This solution follows the notations and rationale taught in the GMATH method. Regards, Fabio. _________________ Fabio Skilnik :: GMATH method creator ( Math for the GMAT) English-speakers :: https://www.gmath.net Portuguese-speakers :: https://www.gmath.com.br ### GMAT/MBA Expert GMAT Instructor Joined 09 Oct 2010 Posted: 1449 messages Followed by: 32 members Upvotes: 59 [quote="fskilnik@GMATH"] AAPL wrote: Veritas Prep Tim and Robert have entered a race, the rules of which stipulate that each runner must run for at least 4 hours and no runner can run for more than 6 hours. Together, they must run a total of 50 miles. If it takes Tim 15 minutes to run a mile and Robert 12 minutes to run a mile, what is the minimum number of miles Robert must run if both Tim and Robert must individually run a whole number of miles? A. 18 B. 20 C. 22 D. 24 E. 26 Alternate approach: (Risky but many-times-"awarded", as in this case!) Let (again) T and R be the number of minutes Tim and Robert run, respectively. To minimize Roberts mileage (our FOCUS), we must minimize R, therefore (by duality) we must maximize T. The maximum POTENTIAL value of T is 6*60 (minutes), hence letÂ´s check whether this number - and R obtained from it - are viable: $${\rm{Tim}}\,\,\left( {6\,\,{\rm{h}}} \right)\,\,:\,\,\,\,6 \cdot 60\,\,\min \,\,\,\left( {{{1\,\,\,{\rm{mile}}} \over {15\,\,\,\min }}\,\matrix{ \nearrow \cr \nearrow \cr } } \right)\,\,\, = \,\,24\,\,{\rm{miles}}$$ $${?_{{\rm{potencial}}\,\left( {{\rm{Robert}}} \right)}}\,\,\,\mathop \Rightarrow \limits^{\sum {\, = \,50\,\,{\rm{miles}}} } \,\,\,\,26\,\,{\rm{miles}}\,\,\,\left( {{{12\,\,\,\min } \over {1\,\,\,{\rm{mile}}}}\,\matrix{ \nearrow \cr \nearrow \cr } } \right)\,\,\, = {2^3} \cdot 3 \cdot 13\,\,\, > \,\,\,\underbrace {{2^3} \cdot 3 \cdot 10}_{4\,\, \cdot \,60}$$ They are! The answer is therefore 26 (miles). Regards, Fabio. _________________ Fabio Skilnik :: GMATH method creator ( Math for the GMAT) English-speakers :: https://www.gmath.net Portuguese-speakers :: https://www.gmath.com.br ### GMAT/MBA Expert GMAT Instructor Joined 25 Apr 2015 Posted: 2456 messages Followed by: 18 members Upvotes: 43 AAPL wrote: Veritas Prep Tim and Robert have entered a race, the rules of which stipulate that each runner must run for at least 4 hours and no runner can run for more than 6 hours. Together, they must run a total of 50 miles. If it takes Tim 15 minutes to run a mile and Robert 12 minutes to run a mile, what is the minimum number of miles Robert must run if both Tim and Robert must individually run a whole number of miles? A. 18 B. 20 C. 22 D. 24 E. 26 Since it takes Tim 15 minutes to run a mile, he runs 4 miles per hour. Similarly, it takes Robert 12 minutes to run a mile, he runs 5 miles per hour. Since each runner must run for at least 4 hours and no runner can run for more than 6 hours, Tim runs at least 4 x 4 = 16 miles and at most 6 x 4 = 24 miles. Similarly, Robert runs at least 4 x 5 = 20 miles and at most 6 x 5 = 30 miles. Since we want to determine the minimum number of miles Robert runs, we can assume Tim runs the the greatest number of miles he possibly can, which is 24. So Robert runs at least 50 - 24 = 26 miles. Answer: E _________________ Scott Woodbury-Stewart Founder and CEO scott@targettestprep.com See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews • 5-Day Free Trial 5-day free, full-access trial TTP Quant Available with Beat the GMAT members only code • 5 Day FREE Trial Study Smarter, Not Harder Available with Beat the GMAT members only code • Magoosh Study with Magoosh GMAT prep Available with Beat the GMAT members only code • FREE GMAT Exam Know how you'd score today for$0

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