2025 AMC 8
Complete problem set with solutions and individual problem pages
Two towns, and , are connected by a straight road, miles long. Traveling from town to town , the speed limit changes every miles: from to to miles per hour (mph). Two cars, one at town and one at town , start moving toward each other at the same time. They drive at exactly the speed limit in each portion of the road. How far from town , in miles, will the two cars meet?
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Solution 1
The first car, moving from town at miles per hour, takes minutes. The second car, traveling another miles from town , takes minutes. The first car has traveled for 3 minutes or th of an hour at miles per hour when the second car has traveled 5 miles. The first car has traveled miles from the previous miles it traveled at miles per hour. They have miles left, and they travel at the same speed, so they meet miles through, so they are miles from town .
Solution 2
From the answer choices, the cars will meet somewhere along the mph stretch. Car travels mph for miles, so we can use dimensional analysis to see that it will be of an hour for this portion. Similarly, car spends of an hour on the mph portion.
Suppose that car travels miles along the mph portion-- then car travels miles along the mph portion. By identical methods, car travels for hours, and car travels for hours.
At their meeting point, cars and will have traveled for the same amount of time, so we have
so , and miles. This means that car will have traveled miles.
