(i) Represent the above information in a Venn-diagram.
(ii) Find the number of students who got 'A' grade in both of these subjects.
(iii) How many students got 'A' grade in maths?

Solution:

Let U be the set of total students. Similarly, let M and E be the sets of students who secured 'A' grade in Mathematics and English, respectively.

Given,
$n(U) = 54$
$n_o(M) = 18$
$n_o(E) = 25$
$n( \overline{M \cup E}) = 7$

To find:
(ii) $n(M \cap E) = ?$
(iii) $n(M) = ?$

For (i)
Show the information in a Venn-diagram.

For (ii)
We know,
$n(U) = n_o(M) + n_o(E) + n(M \cap E) + n( \overline{M \cup E})$

$or, 54 = 18 + 25 + n(M \cap E) + 7$

$or, 54 = 43 + 7 + n(M \cap E)$

$or, 54 = 50 + n(M \cap E)$

$or, 54 - 50 = n(M \cap E)$

$\therefore n(M \cap E) = 4$


For (iii)
We know,
$n(M) = n_o(M) + n(M \cap E)$

$or, n(M) = 18 + 4$

$\therefore n(M) = 22$


Hence,
The required number of students who secured 'A' grade in both of these subjects is 4 students and the total number of students who secured 'A' grade in Mathematics is 22.


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