How many love everything?

The percentages who don't love these activities are 10%, 14%, 20%, and 26%, resp.. If these populations are distinct, then the percentage who loves everything is 100%-10%-14%-20%-26%=30%, or, $\boxed{45}$ students.

Consider the following sets:

A: Students who love singing

B: Students who love dancing

C: Students who love acting

D: Students who love studying

The number of students that do not like all these four activities is given by $|(A \cup B \cup C \cup D)^c| = |(A^c \cap B^c \cap C^c \cap D^c)|$ , where the equality holds due to De'Morgan's Law. An upper bound is given by

$|(A^c \cap B^c \cap C^c \cap D^c)| \leq |A^c| + |B^c| + |C^c| + |D^c|$

$|(A^c \cap B^c \cap C^c \cap D^c)| \leq .1*150 + .14*150 + .2*150 + .26*150$

$|(A^c \cap B^c \cap C^c \cap D^c)| \leq 105$

To find the minimum number of students that love all the activities, we assume that $A^c, B^c, C^c,$ and $D^c$ are disjoint. Consequently, at the most 105 students do not love all these four activities. We conclude that at least 150 - 105 = 45 students love all the activities.

First of all find the percent of all students who loves each things i.e. love singing = 90% so total no. of students who love singing = 90/100 * 150 = 135.

love dancing = 86% so total no. of students who love dancing = 86/100 * 150 = 129.

love acting = 80% so total no. of students who love acting = 80/100 * 150 = 120.

love studying = 74% so total no. of students who love studying =74/100 * 150 =111.

now subtract each of the solution with total i.e. with 150. After subtraction you will get the following solutions...15...21...20...39. now sum up these numbers and subtract form total .....i.e. (150)-(105) = 45 and that's the solution. :)

Minium percentage of student who love...

• Singing and Dancing : $100-(100-90)-(100-86)=76$

• Singing, Dancing and Acting : $100-(100-76)-(100-80)=56$

• Singing, Dancing, Acting and Studying(All) : $100-(100-56)-(100-74)=30$

Hence, the answer is $150.(0.3) =45$