Question

From the students who have participated in an examination, 30% passed in Science, 40% passed in Mathematics and 60% passed in Nepali. Among them 18% passed in Mathematics and Science, 15% passed in Mathematics and Nepali, 20% passed in Nepali and Science, 13% failed in all three subjects and 100 students passed in all three subjects. Then, a) Represent the above information in the form of cardinality. (1) b) Draw a Venn - diagram to illustrate the above information. (1) c) What is the percentage of students who passed in all three subjects? (3) d) Find the number of students who participated in the exam. (1)​

Answer 1

Answer:

(a) See below

(b) See attachment

(c) 10%

(d) 1,000

Step-by-step explanation:

Let S be the set of students who passed in Science.

Le M be the set of students who passed in Mathematics.

Let N be the set of students who passed in Nepali.

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Part (a)

Given that 30% of students passed in Science, 40% passed in Mathematics and 60% passed in Nepali:

$|S| = 0.3$

$|M| = 0.4$

$|N| = 0.6$

Given that 18% of students passed in Mathematics and Science, 15% passed in Mathematics and Nepali, 20% passed in Nepali and Science:

$|M \cap S| = 0.18$

$|M \cap N| = 0.15$

$|N \cap S| = 0.2$

Given that 13% failed in all three subjects:

$|S' \cap M' \cap N'| = 0.13$

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Part (b)

To draw the Venn diagram:

• Draw three overlapping circles to represent the sets Science (S), Mathematics (M), and Nepali (N), and label each circle with the corresponding set name (S, M, N).
• Let x be the percentage of students who passed in all three subjects, so place 'x' in the central section where all three circles overlap.
• Label the overlapping regions with the given percentages minus 'x'.
• Fill in the remaining region of each circle, ensuring the sum of the individual regions equals the percentage given for each subject set.
• Place the percentage of students who failed in all three subjects outside the circles.

See attachment.

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Part (c)

In a Venn diagram, the total percentage of all the regions in the diagram should sum to 100%. Therefore:

$x - 0.08 + 0.07 + x + 0.25 + x + 0.18 - x + x + 0.2 - x + 0.15 - x+0.13=1\\\\x + x + x + x - x - x - x - 0.08 + 0.07 + 0.25 + 0.18 + 0.2 + 0.15+0.13 =1\\\\4x - 3x +0.9=1\\\\x+0.9=1\\\\x=0.1$

So, the percentage of students who passed in all three subjects is 10%.

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Part (d)

To find the total number of students who participated in the exam, we need to divide the number of students who passed in all three subjects (100) by the percentage of students who passed in all three subjects (10%).

$\textsf{Total number of students} = \dfrac{\textsf{Number of students who passed in all three subjects}}{\textsf{Percentage of students who passed in all three subjects}}\\\\\\\textsf{Total number of students} = \dfrac{100}{0.1}\\\\\\\textsf{Total number of students} = 1000$

Therefore, there were 1,000 students who participated in the exam.