In an examination 45% students passed in Science only, 25% passed in English only and 5% failed in both subjects. If 200 students passed in English, find the total number of students.

Answer: n(U) = 400

Solution:

Let U be the set of total students appeared in the examination.

Let S and E be the sets of students who got grade 'A' in Science and English, respectively.

Given,

$n(U) = 100%$

$n_o(S) = 45%$

$n_o(E) = 25%$

$n(\overline{A \cup B}) = 5%$

$n(E) = 200$

Using formula,

$n(U) = n_o(S) + n_o(E) + n(S \cap E) + n(\overline{S \cup E})$

$or, 100% = 45% + 25% + n(S \cap E) + 5%$

$or, 100% = 75% + n(S \cap E)$

$or, n(S \cap E) = 25%$

And,

$n(E) = n_o(E) + n(S \cap E)$

$or, n(E) = 25% + 25%$

$\therefore n(E) = 50%$

Also,

$n(E) = 200$

$or, 50 of n(U) = 200$

$\therefore n(U) = 400$

Hence, the required total number of students who appeared in the examination and were surveyed is 400.

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