Question

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

• A. $510$
• B. $504$
• C. $205$
• D. $405$

Answer 1

The correct option is B $504$

The venn diagram shows the number of rivers polluted by different means. There are total three categories represented by three circles $S$, $P$ and $C$.
Given that total number of rivers polluted by Sulphur were $520$
$S=520$
In the figure $S$ includes $e,a,b$ and $d$.
Hence $e+a+b+d=520$....(1)
Similarly it is given that $P=b+d+f+c=335$....(2)
Also, $C=a+d+c+g=425$.....(3)
Also, it is given that 100 rivers are polluted by both sulphur and crude oil.
Hence from the figure, $a+d=100$....(4)
Similarly $b+d=180$.....(5)
$c+d=150$......(6)
$d=28$......(7) (polluted by all three compounds)
We want to find the number of rivers polluted exactly by any one compound.
That means in the figure we want to find, $e+f+g$.
From (1), (2) and (3) we get $e+f+g+2(a+b+c)+3d=520+335+425=1280$.....(8)
Similarly from (4), (5) and (6) we get $(a+b+c)+3d=430$
Multiplying by 2, we get $2(a+b+c)+6d=860$.....(9)
$3\times$ (7) gives
$2(a+b+c)+3d=860-84=776$......(10)
Now (8) - (10) gives $e+f+g=1280-776=504$