A problem based on set-theory In an examination, at least 70% of the students failed in physics, at least 72% failed in chemistry, at least 80% failed in mathematics and at least 85% failed in english. How many at least must have failed in all the four subjects.(all information is sufficient to calculate the answer)
Well, I don't understand what does that 'at least failed' means. And how would you know  the exact number of students failing/passing in each subject and then calculating the students failing in all four from venn diagram which would be theoretically very difficult. Please help me!
 A: Given 
Number of students failed in Physics $=$ at least $70$% 
Number of students failed in Chemistry $=$ at least $72$% 
Number of students failed in Mathematics $=$ at least $80$% 
Number of students failed in English $=$ at least $85$% 
Number of students passed in Physics $=$ at most $30$% 
Number of students failed in Chemistry $=$ at most $28$%
Number of students failed in Mathematics $=$ at most $20$%
Number of students failed in English $=$ at most $15$%
Therefore, the number of students passed all the given subjects $=$ at most $(30+28+20+15)$% $=$ at most $93$%
Therefore, the number of students failed in all the four subjects $=$ at least $100$%$-93$% $=$ at least $7$%
A: Hint : Take the worst case (minimum case) and try to solve it by Venn Diagram.
A: Hint


*

*If $70\%$ fails Physics and $72\%$ fails Chemistry then, at least, $42\%$ fails both. Don't they?

*If $42\%$ fails Physic and Chemistry and $80\%$ fails Maths then, at least, $22\%$ fails the tree.


Can you get the answer now?
A: Given: $$\begin{cases}P(Ph)\ge 0.7,\\ P(Ch)\ge 0.72,\\P(M)\ge 0.8,\\ P(E)\ge 0.85 \end{cases}$$
we get:
$$P(Ph\cap Ch)=P(Ph)+P(Ch)-P(Ph\cup Ch)\ge P(Ph)+P(Ch)-1\ge 0.42;\\
P(M\cap E)=P(M)+P(E)-P(M\cup E)\ge P(M)+P(E)-1\ge 0.65;\\
P((Ph\cap Ch)\cap(M\cap E))=P(Ph\cap Ch)+P(M\cap E)-P(Ph\cup Ch\cup M\cup E)=\\ P(Ph\cap Ch)+P(M\cap E)-1\ge 0.42+0.65-1=0.07. $$
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