Problem:
Two jars each contain the same number of marbles, and every marble is either blue or green. In Jar the ratio of blue to green marbles is , and the ratio of blue to green marbles in Jar is . There are green marbles in all. How many more blue marbles are in Jar than in Jar ?
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Solution:
Let be the number of green marbles in Jar . Then there are green marbles in Jar . Jar contains blue marbles and marbles in all, and Jar contains blue marbles and marbles in all. Because the jars contain the same number of marbles, , and this equation has the solution . Therefore Jar contains blue marbles, and Jar contains blue marbles. Jar contains more blue marbles than does Jar .
The number of marbles in each jar must be a multiple of both and , so it is a multiple of . If there are marbles in each jar, then there are green marbles in Jar and green marbles in Jar . Thus , so . Therefore there are fewer green marbles in Jar than in Jar , so there are also more blue marbles in Jar than Jar .
The problems on this page are the property of the MAA's American Mathematics Competitions