In a school of 300 students,110 offered french,110 Hausa language, 180 history, 40 french and Hausa, 50 Hausa and history, 60 french and history, while 30did not offer any of the three subjects.
1) draw a Venn diagram to represents the data
2) find the number of students who offered: 1) all the subjects
2) history alone
To represent this data on a Venn diagram and answer the question, we need to count the students who take different combinations of subjects. First, let's tabulate the data:
- Total number of students: 300
- Students offering French (F): 110
- Students offering Hausa language (H): 110
- Students offering History (His): 180
- Students offering French and Hausa (F ∩ H): 40
- Students offering Hausa and History (H ∩ His): 50
- Students offering French and History (F ∩ His): 60
- Students offering all three subjects (F ∩ H ∩ His): Unknown
- Students not offering any of the three subjects: 30
**Step 1: Drawing the Venn Diagram**
A Venn diagram can be drawn using three intersecting circles within a rectangle. Each circle represents one subject (F, H, His), and their intersections represent students who are taking multiple subjects. The rectangle represents the total number of students (300).
Since we can't graphically draw the Venn diagram in this text format, I will provide you with a description to draw it yourself on paper:
1. Draw a large rectangle and label it 300 for the total number of students.
2. Draw three overlapping circles inside the rectangle and label them F, H, and His for French, Hausa, and History.
3. In the region where all three circles intersect, label it as "x" – this will be the number of students who take all three subjects.
4. In the two-subject intersection areas, we subtract "x" from the known numbers for a combination of two subjects:
- For F ∩ H (French and Hausa), label it 40 - x.
- For H ∩ His (Hausa and History), label it 50 - x.
- For F ∩ His (French and History), label it 60 - x.
**Step 2: Calculate the number of students who took all the subjects**
To find "x" (students taking all three subjects), we look at the total number of students taking each subject, and subtract the numbers taking two subjects and all three subjects:
F = (Students taking F only) + (F ∩ H) - x + (F ∩ His) - x + x
H = (Students taking H only) + (H ∩ F) - x + (H ∩ His) - x + x
His = (Students taking His only) + (His ∩ F) - x + (His ∩ H) - x + x
Given that F = 110, H = 110, His = 180, the equations become:
110 = (Students taking F only) + (40 - x) + (60 - x) + x
110 = (Students taking H only) + (40 - x) + (50 - x) + x
180 = (Students taking His only) + (60 - x) + (50 - x) + x
Solving these simultaneously can be complex. Instead, let's use another method: Count the total number of combinations and subtract from the total:
Total = 300
Not taking any subjects = 30
Total taking at least one subject = 300 - 30 = 270
The total number of students taking at least one subject (270) is the sum of all individual subject numbers, minus double counted students (those appearing in two circles), plus students who are counted three times (those in the intersection of all three circles, "x"):
270 = F + H + His - (F ∩ H + F ∩ His + H ∩ His) + x
270 = 110 + 110 + 180 - (40 + 50 + 60) + x
270 = 400 - 150 + x
270 = 250 + x
x = 270 - 250
x = 20
**Answer 1:** The number of students who offered all the subjects is 20.
**Step 3: Calculate the number of students who took only History**
For those who took History alone, we need to subtract from the total number taking History, the number of students taking History with other subjects, and those taking all three:
Students taking History alone = His - (F ∩ His) - (H ∩ His) - x
Students taking History alone = 180 - 60 - 50 - 20
Students taking History alone = 180 - 130
Students taking History alone = 50
**Answer 2:** The number of students who offered History alone is 50.