Question

If $n(A\cup B\cup C)=100,n\left(A\right)=4x,n\left(B\right)=6x,n\left(C\right)=5x,n(A\cap B)=20,n(B\cap C)=15,n(A\cap C)=25$ and $n(A\cap B\cap C)=10$ , then find the value of $x$.
$\left[5\right]$

Answer 1

Given, $n(A\cup B\cup C)=100,n\left(A\right)=4x,n\left(B\right)=6x,n\left(C\right)=5x,n(A\cap B)=20,n(B\cap C)=15,n(A\cap C)=25andn(A\cap B\cap C)=10$
$\left[1\right]$

For any three finite sets A, B and C,
$n(A\cup B\cup C)=n\left(A\right)+n\left(B\right)+n\left(C\right)–n(A\cap B)–n(B\cap C)–n(C\cap A)+n(A\cap B\cap C)$
$\left[1\right]$
$\therefore 100=4x+6x+5x–20–15–25+10$
$\left[1\right]$
$\Rightarrow 100=15x–60+10$
$\Rightarrow 100=15x–50$
$\Rightarrow 15x=100+50=150$
$\Rightarrow x=10$
$\left[2\right]$