In an examination, 25% of the candidates failed in Mathematics and 12%

In an examination, 25% of the candidates failed in Mathematics and 12% failed in English. If 10% of the candidates failed in both the subjects and 292 candidates passed in both the subjects, which one of the following is the number of total candidates appeared in the examination ?

300
400
460
500
This question was previously asked in
UPSC CAPF – 2017
The correct answer is B) 400.
Let M be the percentage of candidates who failed in Mathematics, E be the percentage who failed in English, and B be the percentage who failed in both.
Given: M = 25%, E = 12%, B = 10%.
The percentage of candidates who failed in at least one subject is given by the formula:
P(M U E) = P(M) + P(E) – P(M ∩ E)
Percentage failed in at least one subject = 25% + 12% – 10% = 37% – 10% = 27%.
The percentage of candidates who passed in both subjects is the remaining percentage:
Percentage passed in both = 100% – Percentage failed in at least one subject = 100% – 27% = 73%.
We are given that 292 candidates passed in both subjects.
Let T be the total number of candidates.
So, 73% of T = 292
(73 / 100) * T = 292
T = (292 * 100) / 73
T = 29200 / 73
Dividing 29200 by 73: 292 / 73 = 4 (since 73 * 4 = 292).
So, 29200 / 73 = 400.
The total number of candidates is 400.
This problem uses the principle of inclusion-exclusion for calculating the union of two sets (failures in Mathematics and English). The complement of the set of people failing in at least one subject is the set of people passing in both subjects.