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.
To answer this question using a Venn diagram, we can start by drawing two overlapping circles. The left circle will represent the projects he wrote, and the right circle will represent the projects he produced. The overlapping region in the middle of the circles will represent the projects where he both wrote and produced.
We know the following information:
- He wrote and produced 4 projects, so we can put this number in the overlapping region.
- He wrote a total of 7 projects, so the number in the left circle should be 7.
- He produced a total of 9 projects, so the number in the right circle should be 9.
Now we can fill in the remaining values.
To find the number of projects he wrote but did not produce (part a), we need to subtract the projects he both wrote and produced from the total number of projects he wrote. So, 7 - 4 = 3. This means he wrote but did not produce 3 projects.
To find the number of projects he produced but did not write (part b), we need to subtract the projects he both wrote and produced from the total number of projects he produced. So, 9 - 4 = 5. This means he produced but did not write 5 projects.
Therefore, the answers are:
(a) He wrote but did not produce 3 projects.
(b) He produced but did not write 5 projects.
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.
To answer this question using a Venn diagram, we need to visualize the information provided.
Let's start by drawing the Venn diagram with two overlapping circles. One circle represents the projects he wrote, and the other circle represents the projects he produced.
In the overlapping area between the two circles, we write the number of projects he both wrote and produced, which is 4.
Next, in the area outside the overlapping part of the circles but inside the "Wrote" circle, we write the number of projects he wrote but did not produce. According to the information provided, he wrote a total of 7 projects, so we write 7 - 4 = 3 in this area.
Similarly, in the area outside the overlapping part of the circles but inside the "Produced" circle, we write the number of projects he produced but did not write. He produced a total of 9 projects, so we write 9 - 4 = 5 in this area.
The Venn diagram looks like this:
```
(Wrote)
/ \
-------------------------
| 4 |
|-----------------|
| 5 |
-----------------------
(Produced)
```
Now we can answer the questions:
(a) The number of projects he wrote but did not produce is 3.
(b) The number of projects he produced but did not write is 5.