Out of $70$ student who appeared for a test in Maths and English, $54$ passed atleast in one subject.If $40$ passed in English and $50$ passed in maths,draw a Venn diagram and find:
i) How many passed in both the subject?
ii) How many passed in maths only?
iii) How many didn't pass in English ?

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Solution

$L e t U = u n i v e r s a l s e t i . e n u m b e r o f s t u d e n t t h o s e w h o a p p e a r e d i n t h e t e s t f o r M a t h a n d E n g l i s h . M = n u m b e r o f s t u d e n t w h o p a s s e d i n M a t h s . E = n u m b e r o f s t u d e n t w h o p a s s e d i n E n g l i s h . (V e n n d i a g r a m i s d r a w n a b o v e) G i v e n, n (M) = 50, n (E) = 40, n (M \cup E) = 54 N o w, w e k n o w t h a t - n (M \cup E) = n (M) + n (E) - n (M \cap E) \Rightarrow 54 = 50 + 40 - n (M \cap E) \Rightarrow n (M \cap E) = 90 - 54 = 36 ∴ N u m b e r o f s t u d e n t w h o p a s s e d i n b o t h M a t h s a n d E n g l i s h = 36 (i) N u m b e r o f s t u d e n t w h o p a s s e d i n b o t h t h e s u b j e c t s = 36 (i i) N u m b e r o f s t u d e n t w h o p a s s e d o n l y i n M a t h s = n (M) - n (M \cap E) = 50 - 36 = 14 (i i i) N u m b e r o f s t u d e n t w h o d i d n^{'} t p a s s e d i n E n g l i s h = U - n (E) = 70 - 40 = 30$

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