2008 iTest Problems/Problem 5

Problem

Jerry recently returned from a trip to South America where he helped two old factories reduce pollution output by installing more modern scrubber equipment. Factory A previously filtered % of pollutants and Factory previously filled % of pollutants. After installing the new scrubber system, both factories now filter % of pollutants.

Jerry explains the level of pollution reduction to Michael, "Factory is the much larger factory. It's four times as large as Factory . Without any filters at all, it would pollute four times as much as Factory . Even with the better pollution filtration system, Factory was polluting nearly three times as much as Factory B."

Assuming the factories are the same in every way except size and previous percentage of pollution filtered, find where is the ratio in lowest terms of volume of pollutants unfiltered from both factories installation of the new scrubber system to the volume of pollutants unfiltered from both factories installation of the new scrubber system.

Solution

Let be the volume of the pollutants unfiltered in factory B. That means the volume of pollutants in factory A is .

The old filter filtered 80% of factory A’s pollutants, so the volume of pollutants of factory A with the old filter is . Similarly, the old filter filtered 72% of factory A’s pollutants, so the volume of pollutants of factory A with the old filter is . The total amount of pollutants under the old filtration system is .

Using the same methodology, filtering 99.5% of the pollutants for both factories means that the volume of pollutants for both factories is . The ratio of the total pollutants before the new filtration to the total pollutants after the new filtration is so