Question

The medical records for a class of 28 children show whether they had previously had measles or mumps. The records show 22 have had measles, 13 have had measles and mumps, and 27 have had measles or mumps. How many children had mumps?

Answer 1

18

Explanation:

This is the kind of question that is often posed when learning about Venn Diagrams and/or the intersection of sets of values.

If we let M stand for the number of children who have measles, and P stand for the number of children who have mumps, then we know the following from reading the problem:

#{: (M, "Measles", 22), (P, "Mumps", ?), (M nn P, "Measles and Mumps", 13), (M uu P, "Measles or Mumps", 27) :}#

The key to solving this is to recognize the use of the union formula:

`A uu B = A + B - (A nn B)`

Essentially, adding A and B together can accidentally overcount if there are any items in both A and B at the same time. This is why after you add A and B together, you subtract whatever are in the overlapped group of A and B.

For our problem:

`M uu P = M + P - (M nn P)`

`27 = 22 + P - 13`

`27 = 9 + P`

`P = 27 - 9 = 18`

There are 18 with Mumps.