80 boys took the GCE examination in the three subjects geography, English and mathematics and none of them failed all three subjects, it was noted that 8 passed English only and 10 passed mathematics only, 7 passed mathematics and geography but not English, 40 passed English and mathematics and 21 passed English and geography, altogether 54 passed English.
(a) Draw a venn diagram to represent this information
(b)from venn diagram, find the number of pupils who
(i) Passed geography only,
(ii) Passed all three subjects
(a) Here is a venn diagram to represent this information:
_________ _______
/ \ / \
| E | | G |
\________/ \__/
| |
| |
___|_________ ___|________
/ M \ / M \
| | | |
\_________/ | \___________/
| |
| |
-----------------
E and G
E represents English, M represents Mathematics, and G represents Geography.
(b) From the venn diagram:
(i) Passed geography only: In the diagram, this is represented by the region labeled "G" only. The number of pupils in this region is 7.
(ii) Passed all three subjects: In the diagram, this is represented by the overlapping region of E, M, and G. The number of pupils in this region is 0, as it was mentioned that none of them failed all three subjects.
(a) The Venn diagram representing the given information is as follows:
```
G
__________|_________
| |
E | X |
|___________|________|
| |
M | X |
|___________|________|
```
In the diagram, E represents English, G represents Geography and M represents Mathematics. The number of students passing in each section is mentioned outside the circles, and the number of students passing in combinations of subjects is denoted by overlapping regions.
(b) From the Venn diagram:
(i) The number of students who passed geography only is shown in the region labeled "G" but not overlapping with any other region. In this case, it is not directly provided, so we need to calculate it.
```
Passed geography only = Total students passing Geography - Students passing Geography and English - Students passing Geography and Math + Students passing all three subjects
Passed geography only = 21 - 0 - 7 + 0
= 14
```
Therefore, the number of pupils who passed geography only is 14.
(ii) The number of students who passed all three subjects is shown in the region where all three circles overlap. In this case, it is not directly provided, so we need to calculate it.
```
Passed all three subjects = Total students passing English - Students passing English and Geography - Students passing English and Math + Students passing Geography and English and Math
Passed all three subjects = 54 - 21 - 40 + 0
= -7 (It is not possible to have negative count, so the number of students who passed all three subjects is 0.)
```
Therefore, the number of pupils who passed all three subjects is 0.
To find the number of pupils who passed geography only and the number of pupils who passed all three subjects, we can use the information given and construct a Venn diagram.
(a) Venn Diagram:
First, draw three overlapping circles to represent the three subjects: geography (G), English (E), and mathematics (M).
(i) Passed English only (8 students):
Label this region as E.
(ii) Passed mathematics only (10 students):
Label this region as M.
(iii) Passed mathematics and geography but not English (7 students):
Label the overlapping region between M and G as MG - E.
(iv) Passed English and mathematics (40 students):
Label the overlapping region between E and M as EM.
(v) Passed English and geography (21 students):
Label the overlapping region between E and G as EG.
(vi) Passed English (54 students):
Label the region that includes E but does not overlap with M or G as E - MG.
Now, add up the student counts for each region:
- E: 8 + 40 + 21 + 54
- M: 10 + 7 + 40 + 54
- G: 7 + 21 + 54
(b) From the Venn diagram:
(i) Passed geography only:
To find the number of pupils who passed geography only, look at the region labeled G, which represents the students who passed geography but not English or mathematics.
(ii) Passed all three subjects:
To find the number of pupils who passed all three subjects, look at the region where all three circles overlap (the center region).
Count the number of students in each region visually and calculate the number for each category based on the Venn diagram.
If you draw the diagram, you can see that since 54 passed English, and 17 passed only Math and Geography, only 9 passed Geography only.
So, if we let
x = english and math only
y = english and geography only
z = all three, then we have
x+z = 40
8+y+z = 21
8+x+y+z = 54
Solve those to find x,y,z and then
(i) already noted to be 9
(ii) 7