Question

In a survey of 99 resorts, it was found that 31 had a spa. 39 had a children's club. 8 had a spa and children's club. 6 had all three features. 57 had a fitness center. 15 had a spa and a fitness center. 21 had a fitness center and children's club. Complete parts a) through e). a) How many of the resorts surveyed had only a spa? (Type a whole number.)

Answer 1

Ok, so

With this situation, we could make a Venn's diagram as follows:

Where S is the spa, C is the children's club and F is the fitness center.

a) Now, we want to know how many of the resorts surveyed only had a spa. For this, we can substract:

`31-(2+6+9)=31-17=14`

b) the number of the resorts that had exactly one of these features is:

Exactly spa is: 31-(2+6+9) = 14.

Exactly fitness center is: 57-(15+6+9) = 57-30 = 27.

Exactly a children's show is: 39-(15+6+2) = 16.

If we sum all these values, we got:

14+27+16 = 57

Therefore, the number of the resorts that had exactly one of these features is 57.

c) We want to know the number of the resorts that had at least one of these features. We can find this, if we first sum the number of the resorts that had exactly one of these features :

14+27+16 = 57.

Now, we're going to sum all the middle values in the diagram:

2+6+9+15 = 32.

Then, 57 + 32 = 89.

Therefore, the number of the resorts that had at least one of these features is 89.

d) the number of the resorts had exactly 2 of these features would be:

2+9+15 = 26.

e) the number of the resorts that had none of these features can be found if we substract the number of total resorts and the number of the resorts that had at least one of these features. This is:

99 - 89 = 10.