Question

In a group of 132 people, 50, 60 and 70 people like three different sweets-Berfi, Jalebi and Rasgulla respectively, The number of people who like all the three sweets is half the number of people who like exactly two sweets. The number of people who like exactly any two out of the three sweets is the same as those who like exactly any other two of the three sweets. The number of people who like the three sweets:

• A. 12
• B. 6
• C. 8
• D. none of these

Answer 1

The correct option is A 12

Sum of all circles = s = I + 2 II + 3 III = 180
Number of people liking at least one sweet = X = I + II + III = 132 [Note - Here, we have assumed that every person likes at least 1 sweet]
(S - X) = II + 2 III = 48
Also, given III = $\frac{II}{2}$
$\therefore$ 2 III + 2 III = 48 $\Rightarrow$ III = 12
Alternatively
$\alpha =a+b+c\beta =x+y+z\gamma =k$
Here $\gamma =\frac{1}{2}\beta \Rightarrow 2\gamma =\beta$
Again x = y = z = p
$\therefore \beta =3p\Rightarrow \gamma =\frac{3}{2}pNow\alpha +2\beta +3\gamma =50+60+70=180\Rightarrow \alpha +4\gamma +3\gamma =180\Rightarrow \alpha +7\gamma =\mathrm{180..........}\left(1\right)Again\alpha +\beta +\gamma =132\alpha +2\gamma +\gamma =132\Rightarrow \alpha +3\gamma =\mathrm{132........}\left(2\right)$
From eq. (1) and (2), we get
$\gamma =12$
Hence 12 people like all 3 sweets.