Out of 120 students appeared in an examination, the number of students who passed in Mathematics only is twice the number of students who passed in Science only. If 50 students passed in both subjects and 40 students failed in both subjects.

1. Find the number of students who passed in Mathematics.
2. Find the number of students who passed in Science.
3. Show the result in a Venn-diagram.

Solution:

Let U be the set of people who appeared in the examination.

Let M and as be the sets of students who passed in Mathematics and Science, respectively.

According to the question,

$n(U) = 120$

$n(M \cap S) = 50$

$n(M \cup S)^c = 40$

$n_o(M):n_o(S) = 2:1$

Let x be the number in the ratio.

$n_o(M):n_o(S) = 2x:x$

$n_o(M) = 2x$

$n_o(S) = x$

Using formula,

$n(U) = n_o(M) + n_o(S) + n(M \cap S) + n(M \cup S)^c$

$or, 120 = 2x + x + 50 + 40$

$or, 120 = 3x + 90$

$or, 30 = 3x$

$\therefore x = 10$

So,

$n_o(M) = 2×10 = 20$

$n_o(S) = 10$

Now,

(i) $n(M) = n_o(M) + n(M \cap S)$

$= 20 + 50$

$= 70$

And,

(ii) $n(S) = n_o(S) + n(M \cap S)$

$= 10 + 50$

$= 60$

Hence,

The required number of students who passed in Mathematics and Science are 70 and 60, respectively.

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