[Math] Solving a word problem with a Venn Diagram

discrete mathematics

Consider the following:

Of the 46 children who participate in a competition, 29 sing, 22 act and 14 paint. Of these, 13 sing and act, 11 act and paint, 7 paint and sing, and 5 sing paint and act.

I am asked to determine

  1. How many children do not participate in any of the three activities mentioned.
  2. How many children only paint
  3. How many children sing and act, but do not also paint.

I started with drawing the diagram, and it is shown below:

enter image description here

I cannot understand how -7 children can only act, and how -7 children can only paint. Either I made a mistake, or I have a gap in my understanding of Venn diagrams.

Can someone please point me in the right direction?

Best Answer

For this problem, you need to start from the most interior category.

So 5 sing paint and act by assumption.

Since 13 sing and act, subtract those who can do all these three, the childen who can sing, act but not paint is 8.

Since 11 act and paint, subtract those who can do all these three, the childen who can act, plain but not sing is 6.

Since 7 paint and sing, subtract those who can do all these three, the childen who can sing, plain but not act is 2.

Since 29 sing, subtract those who can only sing&act, only sing&paint and do all these three, children who can only sing is $29-8-2-5=14$.

Similarly, you can fill in the other blanks.

Related Question