Venn diagrams can often be solved – by writing a number or probability in each part – by labelling a part of the diagram x, and deriving an equation which vcan be solved to find x.
In a class of 24 students, all take one or more of music, art and drama. It is known that 13 take music, 17 take art and 16 take drama.
9 take both music and drama, 11 take both music and art and 10 take both art and drama.
How many students take all three?
Start by drawing a Venn diagram, with sets M, A and D to represent the students who take music, art and drama respectively. Label the intersection
9 take both music and drama
This number includes so write
in the Venn diagram as shown.
Similarly for
and
13 take music so the remaining number in M is
17 take art so the remaining number in A is
All the numbers in the Venn diagram add up to the total number of students, so
x+16=24
Hence x=8