Solving Word Problems With Venn Diagrams Three Sets - Free Printable

Problem Analysis and Solution

We are given data about purchases at a pet store over a four-hour period, categorized into products for dogs, cats, fish, and other pets. The task is to use the information provided and the Venn diagram to answer specific questions. Let's break it down step by step.

#### Given Information:
1. 83 purchased a dog product.
2. 101 purchased a cat product.
3. 22 purchased a fish product.
4. 31 purchased both a dog and a cat product.
5. 8 purchased both a dog and a fish product.
6. 10 purchased both a cat and a fish product.
7. 6 purchased all three (dog, cat, and fish) products.
8. 34 purchased a product for a pet other than a dog, cat, or fish.

#### Venn Diagram Breakdown:
The Venn diagram is divided into several regions:
- \( D \): Dog product only.
- \( C \): Cat product only.
- \( F \): Fish product only.
- \( D \cap C \): Both dog and cat products.
- \( D \cap F \): Both dog and fish products.
- \( C \cap F \): Both cat and fish products.
- \( D \cap C \cap F \): All three products (dog, cat, and fish).
- Outside the circles: Products for other pets.

From the diagram:
- \( D \cap C \cap F = 6 \) (all three products).
- \( D \cap C = 25 \) (both dog and cat, excluding those who also bought fish).
- \( D \cap F = 2 \) (both dog and fish, excluding those who also bought cat).
- \( C \cap F = 4 \) (both cat and fish, excluding those who also bought dog).
- \( D = 50 \) (dog product only).
- \( C = 66 \) (cat product only).
- \( F = 10 \) (fish product only).
- Outside the circles: 34 (other pets).

#### Questions to Solve:

##### a. How many purchases were for a dog product only?
From the Venn diagram, the region labeled \( D \) represents purchases for a dog product only. This value is already given as:
\[
\boxed{50}
\]

##### b. How many purchases were for a cat product only?
From the Venn diagram, the region labeled \( C \) represents purchases for a cat product only. This value is already given as:
\[
\boxed{66}
\]

##### c. How many purchases were for a dog or a fish product?
To find the number of purchases for a dog or a fish product, we need to consider the union of the sets \( D \) and \( F \). Using the principle of inclusion-exclusion:
\[
|D \cup F| = |D| + |F| - |D \cap F|
\]
Where:
- \( |D| = 83 \) (total dog product purchases).
- \( |F| = 22 \) (total fish product purchases).
- \( |D \cap F| = 8 \) (both dog and fish products).

Substitute the values:
\[
|D \cup F| = 83 + 22 - 8 = 97
\]
Thus, the number of purchases for a dog or a fish product is:
\[
\boxed{97}
\]

##### d. How many purchases were there in total?
To find the total number of purchases, we sum up all the regions in the Venn diagram, including the purchases for other pets:
1. Dog product only: \( 50 \)
2. Cat product only: \( 66 \)
3. Fish product only: \( 10 \)
4. Both dog and cat: \( 25 \)
5. Both dog and fish: \( 2 \)
6. Both cat and fish: \( 4 \)
7. All three (dog, cat, and fish): \( 6 \)
8. Other pets: \( 34 \)

Add these values together:
\[
50 + 66 + 10 + 25 + 2 + 4 + 6 + 34 = 197
\]
Thus, the total number of purchases is:
\[
\boxed{197}
\]

Final Answers:

\[
\boxed{50, 66, 97, 197}
\]