In a certain class,22 pupils take one or more of Chemistry, Economics and Government. 12 take Economics, 8 take Government, and 7 take Chemistry. Nobody takes Economics and Chemistry and 4 pupils take Economics and Government.
Draw a venn diagram
How many pupils will take
1. Both Chemistry and Government
2. Government only
To draw the Venn diagram, we start by putting the number of pupils taking each subject in the middle of the appropriate circle:
```
Chemistry
/ \
/ \
8 / \ 4
/ \
Economics ----- Government
\ /
\ /
\ /
7
```
We know that 22 pupils take at least one of these subjects, so we add up the numbers in each circle to get:
```
Chemistry
/ \
/ \
0 / \ 0
/ \
Economics ----- Government
\ /
\ /
\ /
0
```
But we're given additional information: nobody takes Economics and Chemistry. So we can eliminate the overlap between those circles:
```
Chemistry
/ \
/ \
0 / \ 0
/ \
Economics ----- Government
/
/
/
0
```
And we're also given that 4 pupils take Economics and Government:
```
Chemistry
/ \
/ \
0 / \ 0
/ \
Economics ----- Government
/
/
/
4
```
To find the answer to the first question, we need to figure out the overlap between the Chemistry and Government circles. We can do this by subtraction:
- 22 pupils total
- 12 take Economics, leaving 10 who don't
- 8 take Government, leaving 14 who don't
- 7 take Chemistry, leaving 15 who don't
- 4 take Economics and Government
If we add up the number of pupils who take each subject or combination of subjects, we get:
- 10 take Economics only
- 14 take Government only
- 15 take Chemistry only
- 4 take Economics and Government
- ? take Chemistry and Government
We know that the total number of pupils taking Chemistry and/or Government is 22, and we can subtract the four combinations listed above to get:
- 22 - 10 - 14 - 15 - 4 = -21
Wait, what? A negative number? This can happen if there is overlap between the circles that we haven't accounted for. And in fact, we have: the pupils taking Economics and Government are counted twice - once in the Economics circle and once in the Government circle. So we need to add them back in once:
- 22 - 10 - 14 - 15 - 4 + 4 = -17
Again, we have a negative number. This means there is some other overlap that we haven't accounted for. In fact, there is only one possibility: the pupils taking Economics and Chemistry and Government. We don't know how many of these there are, but we can label this area of the Venn diagram as "x", with the understanding that x + 4 is the total number taking all three subjects. So:
- x + 4 = ?
We can't solve for x or ?, but we do know that the total number of pupils taking these three subjects is at least 0 (since we can't have a negative number of pupils). So the answer to the first question is "at least 0" (which we could write as "some" or "unknown").
For the second question, we're asked how many pupils take Government only. We can see from the Venn diagram that this is just the number in the Government circle that's not in any of the other circles, which is:
- 14 take Government only
So the answer to the second question is 14.
To draw a Venn diagram for this problem, we start by creating three circles that represent Chemistry (C), Economics (E), and Government (G).
According to the information given:
- 12 students take Economics (E)
- 8 students take Government (G)
- 7 students take Chemistry (C)
To find the number of students who take both Chemistry and Government (C ∩ G), we will subtract the number of students who take Economics and Government (E ∩ G) from the number of students who take Government (G).
1. Both Chemistry and Government (C ∩ G):
To find this number, we subtract the number of students who take Economics and Government (E ∩ G) from the number of students who take Government (G):
C ∩ G = G - E ∩ G = 8 - 4 = 4
2. Government only (G):
To find the number of students who take Government only, we subtract the number of students who take Economics and Government (E ∩ G) and the number of students who take both Chemistry and Government (C ∩ G) from the number of students who take Government (G):
G = G - C ∩ G - E ∩ G = 8 - 4 - 4 = 0
Therefore, no students take Government only.