Question

A bass guitarist worked on 12 music projects last year. He wrote and produced 4 projects. He wrote a total of 7 projects. He produced a total of 9 projects. Use a Venn diagram to determine​ (a) how many projects he wrote but did not​ produce, and​ (b) how many projects he produced but did not write.

Answer 1

To solve this problem using a Venn diagram, we can create two overlapping circles: one for writing and one for producing.

Let's label the writing circle as A and the producing circle as B.

Given that the bass guitarist wrote a total of 7 projects and produced a total of 9 projects, we can write the following information in the Venn diagram:

-A (writing) has 7 projects.
-B (producing) has 9 projects.
The overlap of A and B represents the projects that the bass guitarist both wrote and produced.

Now, we know that the bass guitarist worked on a total of 12 projects. Let's denote the universal set (the set of all projects) as U. The number of elements in U (12 projects) can be calculated by adding the elements in A, B, and the overlap of A and B. So:
U = A + B + (Overlap of A and B)
12 = 7 + 9 + (Overlap of A and B)

Rearrange the equation above to find the overlap of A and B:
Overlap of A and B = 12 - (7 + 9)
Overlap of A and B = 12 - 16
Overlap of A and B = -4

Since -4 projects are not possible, we can conclude that there was an error in the given information. The bass guitarist cannot have written and produced more projects than the total number of projects he worked on.

Thus, we cannot determine the specific numbers of projects that he wrote but did not produce (a) or the number of projects he produced but did not write (b) using the given information and Venn diagram.