Venn diagram story problem

Question

The new owner of a student apartment building near campus was considering a No Pets rule. He decided to survey the current residents to see how many owned pets and what kind of pets they owned. The building had 58 units. The residents reported that 30 owned birds; 38 owned fish; and 30 owned frogs, lizards or snakes. The owner found that 18 had both fish and birds and 15 kept a bird anf frog, lizard or snake. Lastly he determined that 10 owned a fish, a bird, and a frog, lizard, or snake. How many residents could have reported no pets?

Answer 1

Assuming one resident only in each unit, there are 58 residents in the survey. Represent this as a large circle R containing three smaller interlocking circles we'll call B (bird owners), F (fish owners) and other pets (P). Circle R contains 8 zones (Zn):

1. ​Only birds
2. Only fish
3. Only other pets (excluding birds and fish)
4. Birds, fish but no other pets
5. Birds, other pets, but no fish
6. Fish, other pets, but no birds
7. Birds, fish and other pets (10)
8. No pets

The Venn diagram can show these zones. Z1 is the part of the B circle that has no intersections with circles F and P; similarly Z2 for the F circle and Z3 for the P circle. Where circles B, P and F overlap showing elements common to between each set, there are 3 zones trapped within each circle. You should be able to see that Z7 appears in each circle because it represents the set of residents having birds, fish and other pets (frogs, lizards, snakes), while the other zones Z4, Z5 and Z6 have only elements common to two sets. Z8 is the zone outside of all the interlocking circles inside the big circle R. This picture will help you to see what's going on in the equations, which determine how many residents, if any, are in each zone.

B=30=Z1+Z4+Z5+Z7; F=38=Z2+Z4+Z6+Z7; P=30=Z3+Z5+Z6+Z7; R=58=Z1+...+Z8.

18=Z4+Z7; 15=Z5+Z7. But Z7=10, so Z4=8; Z5=5. From this we can find Z1=30-(8+5+10)=7.

Z2+Z6=38-18=20; Z3+Z6=30-15=15; 58=7+(20-Z6)+(15-Z6)+8+5+Z6+10, so 58=65-Z6 and Z6=65-58=7. Now we can find Z2=13; Z3=8; Z8=58-(7+13+8+8+5+7+10)=0. There are no reports of any resident having no pets, so all residents have at least one pet.