Problem:
The students in Mr. Neatkin's class took a penmanship test. Two-thirds of the boys and of the girls passed the test, and an equal number of boys and girls passed the test. What is the minimum possible number of students in the class?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Because of the boys passed, the number of boys in the class is a multiple of . Because of the girls passed, the number of girls in the class is a multiple of . Set up a chart and compare the number of boys who passed with the number of girls who passed to find when they are equal.
The first time the number of boys who passed equals the number of girls who passed is when they are both . The minimum possible number of students is .
Because of the boys passed, the number of boys who passed must be a multiple of . Because of the girls passed, the number of girls who passed must be a multiple of . Because the same number of boys and girls passed, the smallest possible number is , the least common multiple of and . If of boys and of girls passed, there are students in the class, and that is the minimum number possible.
Let the number of girls and the number of boys. Then , so . Because and are relatively prime, the minimum number of boys and girls is boys and girls, for a total of students.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions