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2005 AMC 12A Problem 6

Problem 6 of 25EasierAlgebra

Josh and Mike live 1313 miles apart. Yesterday Josh started to ride his bicycle toward Mike’s house. A little later Mike started to ride his bicycle toward Josh’s house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mike’s rate. How many miles had Mike ridden when they met?

Answer choices

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Solution

Since distance is rate times time, Josh rode 45⋅2=85\dfrac{4}{5} \cdot 2 = \dfrac{8}{5} as far as Mike. Let mm be the miles Mike rode. Then 13=m+85m=135m, 13 = m + \dfrac{8}{5}m = \dfrac{13}{5}m, so m=5.m = 5. Thus, the correct answer is B.

More practice

Concepts: distance rate and time · ratio and proportion

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.