Out of some students who appeared in an examination, 80% passed in Mathematics, 75% passed in English and 5% failed in both subjects. If 300 of them were passed in both subjects, how many students were appeared in the examination?

Answer: 500 students

Solution:

Let U be the set of those students who appeared in the examination and were surveyed.

Let M and E be the sets of students who passed in Mathematics and English, respectively.

Given,

$n(U) = 100%$

$n(M) = 80%$

$n(E) = 75%$

$n(\overline{M \cup E}) = 5%$

$n(M \cap E) = 300$

Using formula,

$n(U) = n(M) + n(E) - n(M \cap E) + n(\overline{M \cup E})$

$or, 100% = 80% + 75% - n(M \cap E) + 5%$

$or, n(M \cap E) = 160% - 100%$

$\therefore n(M \cap E) = 60%$

We know,

$n(M \cap E) = 55% of n(U)$

$or, 300 = \frac{60}{100} × n(U)$

$or, \dfrac{300 × 100}{60} = n(U)$

$\therefore n(U) = 500$

Hence, the required number of students whose results were surveyed is 500.

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