Question

When the members of a family discussed where their annual reunion should take place, they found that out of all the family members, 10 would not go to a park, 9 would not go to a beach, 11 would not go to the family cottage, 3 would go to neither a park nor a beach, 4 would go to neither a beach nor the family cottage, 6 would go to neither a park nor the family cottage, 1 would not go to apark or a beach or to the family cottage,and 2 would go to all three places. What is the total number of family members?

Answer 1

20

Explanation:

Let:

• NP = The non-park goers.

,

• NB = The non-beach goers.

,

• NC = The non-cottage goers.

The Venn diagram below is used to represent the given information:

Given:

• There are 10 non-park goers: a+b+c+g=10

,

• There are 9 non-beach goers: b+d+e+g=9

,

• There are 11 non-cottages goers: c+e+f+g=11

,

• There are 3 non-park and non-beach goers: b+g=3

,

• There are 4 non-beach and non-cottage goers: e+g = 4

,

• There are 6 non-park and non-cottage goers: c+g=6

,

• There is 1 non-park, non-beach, and non-cottage goer: g=1

,

• There are 2 who are neither a non-park, non-beach, or non-cottage goer: h=2

So, the total number of family members will be:

`Total=a+b+c+d+e+f+g+h`

Since g=1:

Next:

`\begin{gathered} c+e+f+g=11 \\ 5+3+f+1=11 \\ f+9=11 \\ f=11-9 \\ f=2 \end{gathered}`

Next:

`\begin{gathered} b+d+e+g=9 \\ 2+d+3+1=9 \\ d+6=9 \\ d=9-6 \\ d=3 \end{gathered}`

Therefore:

`\begin{gathered} Total=(a+b+c+g)+d+e+f+h \\ =10+3+3+2+2 \\ =20 \end{gathered}`

The total number of family members is 20.