Mock AIME 6 2006-2007 Problems/Problem 3

Problem

Alvin, Simon, and Theodore are running around a $1000$ -meter circular track starting at different positions. Alvin is running in the opposite direction of Simon and Theodore. He is also the fastest, running twice as fast as Simon and three times as fast as Theodore. If Alvin meets Simon for the first time after running $312$ meters, and Simon meets Theodore for the first time after running $2526$ meters, how far apart along the track (shorter distance) did Alvin and Theodore meet?

Solution

Alvin and Simon start $\frac{2+1}{2}(312)=468$ metres apart, as they are travelling in different directions. Simon and Theodore start $\frac{3-2}{3}(2526)=842$ metres apart, as they are travelling in the same direction. Therefore, Alvin and Simon are $842-468=\framebox{374}$ metres apart, as they are travelling in different directions. ~AbbyWong

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Mock AIME 6 2006-2007 Problems/Problem 3

Problem

Solution