Question

in a class of 38 students each student studies at least one of the following French mathematics and history. The following table gives further details of the subject studied French only:9 mathematics only :8 history only:1 French and mathematics :7 French and history :9 history and mathematics :8 if x is the number of students who study all the three subjects,
(i) illustrate the above information on the venn diagram
(ii) write an equation in x involving the number of students in the regions of the diagram
(iii) find the number of students who study (i) all the three subjects (ii) French

Answer 1

(i) Below is a Venn diagram representing the above information:

```
F
F ∩ M ∩ H
/ \
F ∩ M F ∩ H
/ \
F only M only
```

(ii) Let's write an equation involving the number of students in the regions of the diagram:
The total number of students who study French is given by:
French only + French and Mathematics + French and History + French, Mathematics, and History
So, we have:
x + 7 + 9 + x = French only + French and Mathematics + French and History + French, Mathematics, and History

(iii) (i) To find the number of students who study all three subjects, we can use the equation from part (ii):
2x + 16 = French only + French and Mathematics + French and History + French, Mathematics, and History
Since there is no given value for French and Mathematics, French and History, or French only, we cannot determine the specific value of x. We can only say that the number of students who study all three subjects is equal to 2x + 16.

(iii) (ii) To find the number of students who study French, we can add up the values from the Venn diagram:
French only + French and Mathematics + French and History + French, Mathematics, and History
x + 7 + 9 + x = 2x + 16
Let's solve for x:
2x + 16 = 2x + 16
The equation simplifies to 0 = 0, which means that we cannot determine the specific number of students who study French.