Chapter 2.2, Problem 20P

A county has two elementary schools, both of which have an after-school program for the convenience of working parents. In school A, $\frac{1}{3}$ of the children attend the after-school program; in school B, $\frac{2}{3}$ of the children attend the after- school program. There are a total of 1500 children in the two schools combined.

a. If school A has 600 students and school B has 900 students, then what is the fraction of elementary school students in the county who attend the after-school program? What if it’s the other way around, and school A has 900 students and school B has 600 students?

b. Using the data in the problem statement, make up two more examples of numbers of students in each school. (These don’t have to be entirely realistic numbers.) For each of your examples. determine the fraction of elementary school children in the county who attend the after-school program.

c. Jamie says that if $\frac{1}{3}$ of the children in school A attend the after-school program, and if $\frac{2}{3}$ of the children from the two schools combined attend the after-school program, then $\frac{1}{2}$ of the children from the two schools combined attend the after-school program. Jamie arrives at this answer because she says $\frac{1}{2}$ is halfway between $\frac{1}{3}$ and $\frac{2}{3}$ . Is Jamine’s answer of $\frac{1}{2}$ always correct? Is Jamie’s answer of $\frac{1}{2}$ ever correct, and if so, under what circumstances?