Problem: In honor of the Fourth of July, a fashion designer was contracted to design and create 5-shirts. Of all the t-shirts the designer creates, 62 of them are red, 88 t-shirts have a flag design, and 86 of them have fireworks on the back of the shirt. Thirteen of these shirts will be red and have fireworks, 19 will have a flag design with fireworks on the back, and twelve will be red with a flag design. There will also be fifteen shirts that have all three characteristics: red with a flag design and fireworks on the back. How many t-shirts does the designer make to meet the demand of the celebration?

Strategy: Create a picture or diagram to organize data.

Solution:

1) First create an organizational tool for data. Since there are three primary styles, a three-way venn diagram is useful.

2) Fill in known data (in boxes on the diagram). The question is how many were made. The designs overlap one another, so it is not as simple as adding 62 + 88 + 86. Instead, totals of each design must be found, then added.

3) Determine number of shirts with only one design: red only, flag design only, or fireworks on the back only. Use known data to find the answer:

• 62 red shirts, but 12 have flags, 15 have all three, 13 have fireworks on the back.
                      62 - 12 - 15 -13 = 22

• 88 flag designs, but 12 are red, 15 have all three, 19 have fireworks on the back.
                      88 - 12 - 15 - 19 = 42

• 86 fireworks on the back, but 13 are red, 15 have all three, 19 have fireworks on the back.
                      86 - 13 - 15 - 19 = 39

4) Add the total of each design, in boxes on the diagram:

• 22 red only
• 12 red with flag
• 13 red with fireworks
• 42 flag design only
• 19 flag and fireworks
• 39 fireworks only
• 15 all three designs

+_______________

162 total shirts designed