Each of the 56 student in the fourth year of a small school studies at least one of the subject history, English and agriculture. Of 14 student who study agriculture, 4 also study history and English, 3 study history not English and 5 study English but not history. Of the 42 student who do not study agriculture, 6 study both history and English, X study only history and 2x study only English. Draw a venn diagram and show the number of student in each subset.find the a. The value of X b. The total number student study English

Answer 1

To solve this problem, we will start by drawing a Venn diagram to represent the information given:

Let's assign the sets as follows:
- H: set of students studying History
- E: set of students studying English
- A: set of students studying Agriculture

From the information given:
- We know that 14 students study Agriculture, so A = 14.
- Of these 14 students, 4 study both History and English, so the intersection of H and E, denoted as H ∩ E, is 4.
- 3 students study History but not English, so H - H ∩ E is 3.
- 5 students study English but not History, so E - H ∩ E is 5.

Next, let's determine the number of students who study only History and only English:

- Of the 42 students who do not study Agriculture, 6 study both History and English, so H - H ∩ E - H ∩ A is 6. (Note that we subtract H ∩ E because we already included those students in H - H ∩ E.)
- Let's define X as the number of students who study only History, so H - H ∩ E - H ∩ A - X represents the number of students who study only English. Since it is given that twice the number of students who study only History is the number of students who study only English, we have X = 2(H ∩ E - H ∩ E - H ∩ A) = 2(4 - 6) = -4.

Now, let's complete the Venn diagram with the information found:

We start by placing the value of X = -4 in the region outside of the three sets H, E, and A as it is negative (indicating that there are no students with this label). Next, we label the regions with the given values.

Let's label the Venn diagram:

```
________
| H |
| _|___
| \ H ∩ E
| _|___
| E |
|_______|
| A |
```

- We know that A = 14.
- We know that H - H ∩ E = 3.
- We know that H - H ∩ E - H ∩ A = 6.
- We know that E - H ∩ E = 5.
- We know that H ∩ E = 4.

Since X is negative, it is not possible to assign it to a region in the Venn diagram.

To find the total number of students studying English, we add the following regions:
- X + E - H ∩ E - H ∩ A, which is -4 + 5 + 4 - 6 = -1.
Since we cannot have a negative number of students, we know that there is an error in the given information.

In summary:
a. The value of X is -4.
b. The total number of students studying English is -1, but this is not possible given the given information.