each of 39 of my friends has either dog, cat, or rabbit. Each of 24 of them has a dog, each of 17 has a cat and each of 16 has a rabbit. the number having both cat and a dog is 1 more than the number having cat and rabbit. there are 9 who have both a dog and a rabbit while two of them have all the three. how many have both rabbit and cat

The Venn diagram above is split into 8 regions labelled A-H.

Each letter represents the number of people in the group. For example, B, C and D represent the number of people owning only a rabbit, cat or dog respectively.

In this problem the interlocking parts of the central circles represent owners of more than one pet.

B+G+H+E=16 rabbit owners

E+H+F+D=24 dog owners

G+H+F+C=17 cat owners

H=2 cat, dog and rabbit owners

F=cat and dog owners, G=cat and rabbit owners, F=G+1

E=9 dog and rabbit owners

A=0 do not own a dog, cat or rabbit

There is an ambiguity about the 39 pet owners: does this mean they own no more than one of cat, dog or rabbit; or it means they own at least one type of pet? In the first case B+C+D=39; in the second case B+C+D+E+F+G+H=39.

Now we can begin to plug in figures.

B+G=16-2-9=5, F+D=24-2-9=13, and G+1+D=13, so G+D=12

G+H+F+C=17; G+2+G+1+C=17; 2G+C=14

If we assume B+C+D+E+F+G+H=39, then (B+G)+(G+1+D)+(C+9+2)=39

5+13+C+11=39; C=39-(5+13+11)=39-29=10; 2G+C=14, so 2G=14-10=4 and G=2; and so there are 2 cat and rabbit owners. (Also, since B+G=5, B=3 who only own a rabbit; and D=12-2=10 only own a dog.)

answered Apr 18, 2017 by Top Rated User (545,980 points)