If a class of 29 children, each of 20 children has a dog and each of 15 has a cat' How many of the c

1
Tutorial

• If a class of 29 children, each of 20 children
  has a dog and each of 15 has a cat. How many of
  the children have both a dog and cat?
• (A) None of the children necessarily has both.
• (B) Exactly 5
• (C) Exactly 6
• (D) At least 6 and at most 15

Dogs and Cats
2
Tutorial

• If a class of 29 children, each of 20 children
  has a dog and each of 15 has a cat. How many of
  the children have both a dog and cat?
• (A) None of the children necessarily has both.
• (B) Exactly 5
• (C) Exactly 6
• (D) At least 6 and at most 15

• Each of the 29 children in this class can be
  classified as one and only one of the following
• Dog only
• Cat only
• Both Dog and Cat
• Neither Dog nor Cat

Dogs and Cats
3
Tutorial

• If a class of 29 children, each of 20 children
  has a dog and each of 15 has a cat. How many of
  the children have both a dog and cat?
• (A) None of the children necessarily has both.
• (B) Exactly 5
• (C) Exactly 6
• (D) At least 6 and at most 15

• Using this classification we can construct a Venn
  Diagram to help us solve this problem.
• Dog only
• Cat only
• Both Dog and Cat
• Neither Dog nor Cat

Both
Dog
Cat
Neither
Total children
Dogs and Cats
4
Tutorial

• If a class of 29 children, each of 20 children
  has a dog and each of 15 has a cat. How many of
  the children have both a dog and cat?
• (A) None of the children necessarily has both.
• (B) Exactly 5
• (C) Exactly 6
• (D) At least 6 and at most 15

For this problem, it is important that the sum of
the number of children in each of the four
classifications does not exceed 29.
Dogs and Cats
5
Tutorial

• If a class of 29 children, each of 20 children
  has a dog and each of 15 has a cat. How many of
  the children have both a dog and cat?
• (A) None of the children necessarily has both.
• (B) Exactly 5
• (C) Exactly 6
• (D) At least 6 and at most 15

If 20 children have a dog only and 15 children
have a cat only, then based this scenario there
would be already be 35 total children in the
class. 20 15 35 Therefore, at least 6
children must have both a dog and a cat. 35 29
6
6
20
15
35
Dogs and Cats
6
Tutorial

• If a class of 29 children, each of 20 children
  has a dog and each of 15 has a cat. How many of
  the children have both a dog and cat?
• (A) None of the children necessarily has both.
• (B) Exactly 5
• (C) Exactly 6
• (D) At least 6 and at most 15

If six children have both a dog and a cat, this
means 14 children have a dog only and 9 children
have a cat only.
6
14
9
Dogs and Cats
7
Tutorial

• If a class of 29 children, each of 20 children
  has a dog and each of 15 has a cat. How many of
  the children have both a dog and cat?
• (A) None of the children necessarily has both.
• (B) Exactly 5
• (C) Exactly 6
• (D) At least 6 and at most 15

In this scenario, a total of 29 children are
accounted for. 14 11 6 29 This means that
there are no children who have neither a dog nor
a cat.
6
14
11
0
29
Dogs and Cats
8
Tutorial

• If a class of 29 children, each of 20 children
  has a dog and each of 15 has a cat. How many of
  the children have both a dog and cat?
• (A) None of the children necessarily has both.
• (B) Exactly 5
• (C) Exactly 6
• (D) At least 6 and at most 15

By changing the number of children who have both
a dog and a cat, we can see that this number can
increase from 6.
Dogs and Cats
9
Tutorial

• If a class of 29 children, each of 20 children
  has a dog and each of 15 has a cat. How many of
  the children have both a dog and cat?
• (A) None of the children necessarily has both.
• (B) Exactly 5
• (C) Exactly 6
• (D) At least 6 and at most 15

For example, a total of 7 children can have both
a dog and a cat. 13 8 7 1 29
7
13
8
1
29
Dogs and Cats
10
Tutorial

• If a class of 29 children, each of 20 children
  has a dog and each of 15 has a cat. How many of
  the children have both a dog and cat?
• (A) None of the children necessarily has both.
• (B) Exactly 5
• (C) Exactly 6
• (D) At least 6 and at most 15

As another example, a total of 8 children can
have both a dog and a cat. 12 7 8 2 29
8
12
7
2
29
Dogs and Cats
11
Tutorial

• If a class of 29 children, each of 20 children
  has a dog and each of 15 has a cat. How many of
  the children have both a dog and cat?
• (A) None of the children necessarily has both.
• (B) Exactly 5
• (C) Exactly 6
• (D) At least 6 and at most 15

Since 15 students have a cat, the maximum number
of children who can have both a dog and a cat is
15 children. That is, all children who have a
cat also have a dog.
Dogs and Cats
12
Tutorial

• If a class of 29 children, each of 20 children
  has a dog and each of 15 has a cat. How many of
  the children have both a dog and cat?
• (A) None of the children necessarily has both.
• (B) Exactly 5
• (C) Exactly 6
• (D) At least 6 and at most 15

Returning to the original problem, the number of
children who have both a dog and a cat must
be (D) At least 6 and at most 15
Dogs and Cats