### Let's Think About It

Mr. Tristany counts all of the boys and girls in the 7th grade at his school. Field Day is quickly approaching and Mr. Tristany is in charge of dividing some of the students into teams that are equally matched. All of the students will not be on teams, but for those that are, each team must have the same number of boys and the same number of girls as all the other teams. Mr. Tristany wants to make sure that there are enough students to make at least 10 teams so he decides to calculate the greatest number of teams that can be formed. He uses a Venn Diagram to make his calculation. The diagram that he used is below.

What is the greatest number of teams that can be formed with the given number of boys and girls?

In this concept, you will learn how to draw a Venn diagram to organize materials.

### Guidance

A **Venn diagram** shows the common numbers in two sets of objects or numbers by using overlapping circles. A Venn diagram helps you to organize data in a visual way to notice patterns and solve for the answer.

There are 280 girls and 260 boys playing on soccer teams this fall. If each team has the same number of girls and the same number of boys, what is the greatest number of teams that can be formed?

A Venn diagram is used to show things that are common and things that are different. For this example, you can write the prime factors of 280 in one circle, the prime factors of 260 in the other circle and the common factors in the part of the circle that overlaps.

By looking at this diagram, you can see that the common factors between 280 and 260 are 5, 2 and 2. If you multiply these together, you will have the total number of groups possible.

\begin{align*}5\times 2\times 2=20\end{align*}

There are 20 possible groups.

### Guided Practice

What is the greatest common factor of 66 and 72?

First, identify the common factors of 72 and 66.

3 and 2

Then, multiply those numbers to calculate the greatest common multiple.

\begin{align*}3\times 2=6\end{align*}

The answer is 6.

### Examples

#### Example 1

Use the Venn diagram to write the greatest common multiple of 225 and 90.

First, identify the common factors of 225 and 90.

5, 3 and 3

Then, multiply those numbers to calculate the greatest common multiple.

\begin{align*}5\times 3\times 3=45\end{align*}

The answer is 45.

#### Example 2

What is the greatest common factor of 98 and 44?

First, identify the common factors.

2

Then, identify the greatest common factor.

2

The answer is 2.

#### Example 3

What values should be listed in the middle section of the Venn diagram below?

First, identify the value of the factors on one of the sides.

\begin{align*}7\times 2\times 2=28\end{align*}

Next, identify the factors that are missing from the 112 side.

2 and 2

Then, write those factors in the middle section.

2 and 2

The answer is 2 and 2.

### Follow Up

Remember Mr. Tristany and his plans for Field Day? He counts all of the boys and girls in the 7th grade at his school. He wants the same number of boys and girls on each team and he wants to make sure that there are enough students to make at least 10 teams so he decides to calculate the greatest number of teams that can be formed. He uses a Venn diagram to make his calculation. The diagram that he used is below.

What is the greatest number of teams that can be formed with the given number of boys and girls?

First, identify the common factors of 105 and 120.

5 and 3

Then, multiply those numbers to calculate the greatest common multiple.

\begin{align*}5\times 3=15\end{align*}

The answer is 15. Mr. Tristany has enough students to fill 15 teams, which is greater than the 10 teams that he wants to build.

### Video Review

### Explore More

Create a Venn diagram for the following data.

1. The factors of 20 and 30.

2. The factors of 45 and 55

3. The factors of 67 and 17

4. The factors of 54 and 18

5. The factors of 27 and 81

6. The factors of 9 and 18

7. The factors of 100 and 200

8. The factors of 8 and 80

9. The factors of 120 and 144

10. The factors of 120 and 140

11. The factors of 80 and 100

12. The factors of 10 and 60

13. The factors of 70 and 140

14. The factors of 6 and 60

15. The factors of 330 and 900

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 9.22.