2015 AMC 8 Problems/Problem 16
In a middle-school mentoring program, a number of the sixth graders are paired with a ninth-grade student as a buddy. No ninth grader is assigned more than one sixth-grade buddy. If of all the ninth graders are paired with of all the sixth graders, what fraction of the total number of sixth and ninth graders have a buddy?
Let the number of sixth graders be , and the number of ninth graders be . Thus, , which simplifies to . Since we are trying to find the value of , we can just substitute for into the equation. We then get a value of
We see that the minimum number of ninth graders is , because if there are then there is ninth-grader with a buddy, which would mean sixth graders with a buddy, and that's impossible. With ninth-graders, of them are in the buddy program, so there sixth-graders total, two of whom have a buddy. Thus, the desired fraction is .