Question

In a pollution study of $1500$ Indian rivers, the following data were reported. $520$ were polluted by sulphur compounds, $335$ polluted by phosphate, $425$ were polluted by crude oil, $100$ were polluted by both crude oil and sulphur compounds, $180$ were polluted by both sulphur compounds and phosphates, $150$ were polluted by both phosphates and crude oil and $28$ were polluted by sulphur compounds, phosphates, and crude oil. How many of the rivers were polluted by exactly one on the three impurities

Answer 1

Let $S=$ the set of river polluted by sulphur compounds.
$P=$ the set of river polluted by phosphates.
$C=$ the set of river polluted by crude oil.
Then the set of river polluted at least one of the impurities is $S\cup P\cup C$
Now $28$ of the river were polluted by all three of the impurities.
So the number of elements in the region $S\cap P\cap C$ is $28$ as indicated in the venn diagram.
Since we are given that $180$ rivers were polluted by both sulphur and phosphate,
there are $180$ elements in $S\cap P$, but $28$ of them lie in $S\cap P\cap C$ which leaves $180-28=152$
River that were polluted only by phosphate.
Similarly $150-28=122$ were polluted only by phosphate and crude oil
while $100-28=72$ were polluted only by sulphur and crude oil.
From the given data, number of river polluted by sulphur is $520$.
Therefore, the number of elements in the remaining portion of the circle $S$ $520-72-28-152=268$
Similarly, the number of elements in the remaining portion of the circle $C$ is $425-122-28-72=203$

Similarly, the number of elements in the remaining portion of the circle $P$ is $335--122-28-152=33$

Rivers polluted by exactly one pollutant is $268+203+33=504$