2015 AMC 8 Problems/Problem 16
Contents
Problem
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?
Solution
Solution 1
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
Solution 2
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
.