In a class, 40 students passed in Mathematics, 50% of the students passed in English, 5% of the students failed in Mathematics and English, and 25% of the students passed in both the subjects. What is the ratio of the number of students who passed in English to that in Mathematics?
[amp_mcq option1=”1 : 1″ option2=”2 : 3″ option3=”5 : 7″ option4=”10 : 9″ correct=”option3″]
Percentage of students failed in Mathematics and English = 5%.
This means 95% of students passed in at least one subject (Mathematics or English or both).
Percentage of students passed in English = 50%.
Percentage of students passed in both subjects (Mathematics and English) = 25%.
Let P(M) be the set of students who passed in Mathematics, and P(E) be the set of students who passed in English.
We are given:
|P(M ∩ E)| / S = 0.25
|P(E)| / S = 0.50
|P(M ∪ E)| / S = 1 – 0.05 = 0.95
Using the principle of inclusion-exclusion for percentages:
|P(M ∪ E)| / S = |P(M)| / S + |P(E)| / S – |P(M ∩ E)| / S
0.95 = |P(M)| / S + 0.50 – 0.25
0.95 = |P(M)| / S + 0.25
|P(M)| / S = 0.95 – 0.25 = 0.70
So, 70% of the total students passed in Mathematics.
We are given that the number of students who passed in Mathematics is 40.
|P(M)| = 40.
Therefore, 70% of S = 40.
0.70 * S = 40
S = 40 / 0.70 = 400 / 7.
Number of students who passed in English = |P(E)| = 50% of S = 0.50 * S.
|P(E)| = 0.50 * (400 / 7) = 0.5 * 400 / 7 = 200 / 7.
The ratio of the number of students who passed in English to that in Mathematics is:
|P(E)| : |P(M)|
(200 / 7) : 40
To simplify the ratio, divide both numbers by 40:
(200 / 7) / 40 : 40 / 40
(200 / (7 * 40)) : 1
(200 / 280) : 1
(20 / 28) : 1
(5 / 7) : 1
The ratio is 5 : 7.