Question

In a survey of 1000 families, it is found that 484 families use electric stoves, 552 families use gas stoves. If all the families use atleast one of these two types of stoves, find how many families use both the stoves?

Answer 1

Let E denote the set of families using electric stove and G denote the set of families using gas stove. Then $n\left(E\right)=484,n\left(G\right)=552,n(E\cup G)=1000$. Let $x$ be the number of families using both the stoves. Then $n(E\cap G)=x$.
Using the results
$n(E\cup G)=n\left(E\right)+n\left(G\right)-n(E\cap G)$
$1000=484+552-x$
$\Rightarrow x=1036-1000=36$
Hence, 36 families use both the stoves.
Aliter
From the Venn diagram,
$484-x+x+552-x=1000$
$\Rightarrow 1036-x=1000$
$\Rightarrow -x=-36$
$x=36$
Hence , 36 families use both the stoves.