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?
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.