2017 AMC 10B Problems/Problem 12
Elmer's new car gives percent better fuel efficiency, measured in kilometers per liter, than his old car. However, his new car uses diesel fuel, which is more expensive per liter than the gasoline his old car used. By what percent will Elmer save money if he uses his new car instead of his old car for a long trip?
Suppose that his old car runs at km per liter. Then his new car runs at km per liter, or km per of a liter. Let the cost of the old car's fuel be , so the trip in the old car takes dollars, while the trip in the new car takes . He saves .
Because they do not give you a given amount of distance, we'll just make that distance miles. Then, we find that the new car will use . The old car will use . Thus the answer is .
You can find that the ratio of fuel used by the old car and the new car for the same amount of distance is , and the ratio between the fuel price of these two cars is . Therefore, by multiplying these two ratios, we get that the costs of using these two cars is So the percentage of money saved is .
-Quadraticfunctions (edited by mydad)
Assume WLOG that Elmer's old car's range is miles. So, Elmer's new car's range is miles. Also, assume that the gas Elmer's old car uses is , which means that diesel will cost . Now we can deduce that Elmer's old car uses per mile, and Elmer's new car uses per mile. Therefore, Elmer's new car saves more money than his old car.
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