Venn Diagram

4. A guidance counselor is planning schedules for 30 students. Sixteen students say they want to take French, 16 want to take Spanish, and 11 want to take Latin. Five say they want to take both French and Latin, and of these, 3 wanted to take Spanish as well. Five want only Latin, and 8 want only Spanish. How many students want French only?

• Start with the Latin circle. You have sufficient information to determine how many students want both Latin and Spanish (1).
  

• Using this new information, you can work within the Spanish circle to determine how many students want both Spanish and French (4).
  

• Using this new information, you can work within the French circle to determine how many students want French only (7).
  

• The last step could be represented algebraically as

x + 4 + 3 + 2 + 5 + 8 + 1 = 30
x + 23 = 30
x = 7

Venn Diagram

3. A veterinarian surveys 26 of his patrons. He discovers that 14 have dogs, 10 have cats, and 5 have fish. Four have dogs and cats, 3 have dogs and fish, and one has a cat and fish. If no one has all three kinds of pets, how many patrons have none of these pets?

7 + 4 + 0 + 3 + 1 + 5 + 1 + x = 26

21 + x = 26

x = 5 patrons have none of these animals

Must be 5 iguana lovers!

Venn Diagram Q

2. In a school of 320 students, 85 students are in the band, 200 students are on sports teams, and 60 students participate in both activities. How many students are involved in either band or sports?

25 + 60 + 140 = 225

There are 225 students involved
in either band or sports.

CAUTION! The fact of "320 total students" mentioned in this problem was not used in the actual solution. Be sure to organize your thinking clearly. A problem may include a piece of information that is not needed to solve the problem. Beware!