Question

At the north campus of a performing arts​ school, ​20% of the students are music majors. At the south​ campus, 70​% of the students are music majors. The campuses are merged into one east campus. If ​30% of the 1000 students at the east campus are music​ majors, how many students did the north and south campuses have before the​ merger?

Answer 1

The north campus had
`800` while the south campus had
`200`.

Explanation:

The question stated that there are
`1000` students in the merged campus. That's the sum of the number of students in the north and the south campus before the merger.

Let
`x` denote the number of students in the north campus before the merger. The other
`(1000 -x)` students would all belong to the south campus before the merger.

Before the merger,
`20\%` of the students of the north campus are of music majors. In terms of
`x`, that's
`(20\%)\cdot x = 0.20\,x` students.

On the other hand,
`70\%` of the students of the south campus are of music majors. That corresponds to
`(70\%)\cdot (1000 - x) = 0.70\cdot (1000 - x) = 700 - 0.70\, x` students.

With a similar logic, the number of music students in the merged east campus will be
`30\% * 1000 = 300`.

The question implies that the sum of students in the two campuses should be equal to the number of music students in the east campus. That is:

`0.20\, x+ (700 - 0.70\, x) = 300`.

Solve for
`x`:

`x = 800`.

In other words, there are
`800` students in the north campus and
`(1000 - 800) = 200` students in the south campus before the merger.