In a school there are $20$ teachers who teach mathematics or physics. Of these, $12$ teach mathematics and $4$ teach both physics and mathematics. How many teach physics ?
Let $M$ denote the set of teachers who teach mathematics and $P$ denote the set of teachers who teach physics. In the statement of the problem, the word 'or' gives us a clue of union and the word 'and' gives us a clue of intersection. We, therefore, have
$n( M \cup P )=20, n( M )=12 \text { and } n( M \cap P )=4$
We wish to determine $n( P ).$
Using the result $n( M \cup P )=n( M )+n( P )-n( M \cap P )$
we obtain $20=12+n(P)-4$
Thus $n( P )=12$
Hence $12$ teachers teach physics.
Out of $800$ boys in a school, $224$ played cricket, $240$ played hockey and $336$ played basketball. Of the total, $64$ played both basketball and hockey; $80$ played cricket and basketball and $40$ played cricket and hockey; $24$ played all the three games. The number of boys who did not play any game is
Let $\mathrm{U}$ be the set of all triangles in a plane. If $\mathrm{A}$ is the set of all triangles with at least one angle different from $60^{\circ},$ what is $\mathrm{A} ^{\prime} ?$
In a certain town, $25\%$ of the families own a phone and $15\%$ own a car; $65\%$ families own neither a phone nor a car and $2,000$ families own both a car and a phone. Consider the following three statements
$(A)\,\,\,5\%$ families own both a car and a phone
$(B)\,\,\,35\%$ families own either a car or a phone
$(C)\,\,\,40,000$ families live in the town
Then,
A survey shows that $63 \%$ of the people in a city read newspaper $A$ whereas $76 \%$ read newspaper $B$. If $x \%$ of the people read both the newspapers, then a possible value of $x$ can be
Two newspaper $A$ and $B$ are published in a city. It is known that $25\%$ of the city populations reads $A$ and $20\%$ reads $B$ while $8\%$ reads both $A$ and $B$. Further, $30\%$ of those who read $A$ but not $B$ look into advertisements and $40\%$ of those who read $B$ but not $A$ also look into advertisements, while $50\%$ of those who read both $A$and $B$ look into advertisements. Then the percentage of the population who look into advertisement is