72% of the students of a certain class took Biology and 44% took Mathematics. If each student took at least one of Biology or Mathematics and 40 students took both of these subjects, the total number of students in the class is:

Let us assume that the N number of people have only TV.
According to question,
? (40 ? P) + P + 10 + 15 + (50 ? P) = 100
? 115 ? 2P + P = 100
? P = 115 ? 100 = 15
? Only TV = 75 ? 15 ? 10 ? 15 = 35

As per given question,
(B ? C) = {b, c, e} ? {b, c, f} = {b, c, e, f}
? A × (B ? C) = {a, d} × {b, c, e, f}
= {(a, b), (a, c), (a, e), (a, f), (d, b), (d, c), (d, e), (d, f)} ?...........(1)
Also, (A × B) = {(a, b), (a, c), (a, e), (d, b), (d, c), (d, e)}
and, (A × C) = {(a, b), (a, c), (a, f), (d, b), (d, c), (d, f)} ?............(2)
? (A × B) ? (A × C) = {(a, b), (a, c), (a, e), (a, f), (d, b), (d, c), (d, e), (d, f)}
From (1) and (2), we have
A × (B ? C) = (A × B) ? (A × C).

Let P = set of people who like coffee and Q = set of people like tea.
Then, P ? Q = set of people who like at least one of the two drinks.
And P ? Q = set of people who like both the drinks.
Given in the question,
n(P) = 37, n(Q) = 52, n(P ? Q) = 70.
Use the below formula,
n(P ? Q) = n(P) + n(Q) ? n(P ? Q)
Put the value from the given question,
70 = 37 + 52 ? n(P ? Q)
? n(P ? Q) = 89 ? 70 = 19.
? 19 people like both coffee and tea.

Let, E be the set of people who are egg-eaters and M be the set of people who are meat-eaters.
As per given question,
n(E) = 3200, n(M) = 2500, n(E ? M) = 1500.
Use the formula
n(E ? M) = n(E) + n(M) ? n(E ? M)
= 3200 + 2500 ? 1500
= 5700 ? 1500 = 4200.
? Number of pure vegetarians = n(U) ? n(E ? M)
Number of pure vegetarians = 5000 ? 4200 = 800.

Given in the question,
n(H ? E) = 1000, n(H) = 750, n(E) = 400
Use the below formula,
n(H ? E) = n(H) + n(E) ? n(H ? E)
We get 1000 = 750 + 400 ? n(H ? E)
? n(H ? E) = 1150 ? 1000 = 150.
Number of People who can speak Hindi only
= n(H ? E?) = n(H) ? n(H ? E)
= 750 ? 150 = 600.

Given in the question,
n(M ? B) = 50, n(M) = 35, n(B) = 37, n(M ? B) =?
use the below formula
n(M ? B) = n(M) + n(B) ? n(M ? B)
We get 50 = 35 + 37 ? n(M ? B)
? n(M ? B) = 35 + 37 ? 50 = 72 ? 50 = 22
? 22 students have opted for both Mathematics and Biology.
Again number of students who have opted for only Mathematics = n(M) ? n(M ? B) = 35 ? 22 = 13.

Area = 1/2 hectare = 10000 / 2 m²
= 5000 m²

Again Area = 1/2 x (Diagonal)²

So 1/2 x (Diagonal)² = 5000m²
? Diagonal²= 10000
? Diagonal = 100

Altitude = ?(13)² - (5)²
= ?144 = 12m

? Area of the triangle = (5 x 12 ) / 2 m²
= 30m²

Let length = l and breadth = b
Then, area = lb

New length = 2l
And new breadth = b/2
? New area = ( 2l ) x (b/2) = lb

So, there is no change in area .

Length of carpet = Total Cost / Rate
= 3600 / 30
= 120 m

Area of carpet = (120 x 75) / 100 m²
= 90 m²
? Area of the room = 90 m²

Breadth of the room = Area /Length
= 90 / 15 m
= 6m