Problem:
Adults made up of the crowd of people at a concert. After a bus carrying more people arrived, adults made up of the people at the concert. Find the minimum number of adults who could have been at the concert after the bus arrived.
Solution:
Let be the number of people at the concert before the bus arrived, and let be the number of adults on the bus. Then the number of adults at the concert before the bus arrived is , and the number of adults at the concert after the bus arrived is . The fraction of adults at the concert after the bus arrived is
and this equation simplifies to . It follows that must be a multiple of 300 , and there is a positive integer such that and . The minimum value of is 1 , corresponding to . This corresponds to . The requested minimum number of adults is .
OR
Let be defined as above. Because and are integers, it follows that 300 divides . Setting shows that the minimum number of adults at the concert is , as above.
The problems on this page are the property of the MAA's American Mathematics Competitions