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
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.
To solve this problem, let's draw a Venn diagram to represent the information given.
Based on the information provided:
- 14 students study agriculture.
- 4 students study both history and English, and also study agriculture.
- 3 students study history but not English and also study agriculture.
- 5 students study English but not history and also study agriculture.
Now let's calculate the values for the Venn diagram:
Let A represent the students studying agriculture.
Let H represent the students studying history.
Let E represent the students studying English.
a) The value of X:
From the given information, we know that 14 students study agriculture and 4 students study both history and English while studying agriculture. Therefore, X = 4.
b) The total number of students studying English:
To determine the total number of students studying English, we need to consider three groups:
1. Students studying only English (represented by E but not H).
2. Students studying both English and history (represented by E and H).
3. Students studying only English and agriculture (represented by E but not H or A).
From the given information, we know that 6 students study both history and English (6 students are represented in the overlap of H and E, which means they are not part of this calculation) and 5 students study English but not history and also study agriculture (they are part of group 3, not part of this calculation).
Assuming there are t total students studying English, the equation will be:
t = (2x) + (6 - 5)
t = 2x + 1
Therefore, the total number of students studying English is given by 2x + 1.
Note: We don't have enough information to determine the precise value of x or the total number of students studying English based on the given information.
To solve this problem, let's break it down step by step:
1. Start by drawing a Venn diagram with three overlapping circles representing history, English, and agriculture. Label each circle accordingly.
2. From the given information, we know that there are a total of 56 students in the fourth year. So, the total number of students is represented by the sum of the subsets in the diagram.
3. Next, we are told that 14 students study agriculture. Write "14" inside the agriculture circle.
4. Of the students who study agriculture, we are also given the following information:
- 4 students study both history and English.
- 3 students study history but not English.
- 5 students study English but not history.
To figure out the number of students in each of these specific subsets, you can use the formula: Total (A) = A only + Both + B only.
- Using this formula, add the given numbers:
- A only + Both + B only = 14
- 3 + 4 + 5 = 12
So, we can conclude that there are 12 students who study either history or English in addition to agriculture.
5. Therefore, the remaining students who do not study agriculture can be determined by subtracting these 12 students from the total number of students (56 - 12 = 44). Represent this number outside the agriculture circle.
6. Moving on to the students who do not study agriculture, we are given the following information:
- 6 students study both history and English.
- X students study only history.
- 2X students study only English.
7. To find the value of X, we can use the given equation: 6 + X + 2X = 44
- Combine like terms: 3X + 6 = 44
- Subtract 6 from both sides of the equation: 3X = 38
- Divide both sides by 3: X = 38/3 = 12.67 (approx)
Since we cannot have a fraction of a student, we can conclude that X is approximately 13.
8. Now that we know the value of X, we can fill in the respective subsets on the Venn diagram:
- Inside the history circle: 3 (history only) + 4 (both) + 13 (X, which is approximately 13) = 20
- Inside the English circle: 5 (English only) + 4 (both) + 26 (2X, which is approximately 26) = 35
The Venn diagram should now accurately represent the given information, and the subsets should each have the appropriate number of students.
In conclusion:
a. The value of X is approximately 13.
b. The total number of students studying English is 35.