AMC 10 Daily Practice Round 3
Complete problem set with solutions and individual problem pages
When two ants work together they can build an anthill in minutes. When the bigger ant works alone, an anthill can be built in minutes less than when the smaller ant works alone. How many minutes does it take the smaller ant to build an anthill when working alone?
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Let be the number of minutes that it would take the bigger ant to build an anthill alone and let be the number of minutes that it would take the smaller ant to build an anthill alone. Since the bigger ant can build an anthill in minutes, the bigger ant builds anthills per minute. Likewise, the smaller ant can build anthills per minute. Thus, working together, the two ants build anthills per minute. It is also given that it takes the two ants minutes to build an anthill together, so this means they build anthills per minute working together. Hence, we get the equation . Multiplying this equation through by gives . From the other given condition, we get , so we can substitute to get . Expanding and rearranging, this equation becomes , which can be factored as . If , then , which does not make sense since must be positive. Therefore, .
