Competition Mode Question

In a multiplex 3 shows A, B and C run in 3 auditoriums. One day, 1000 visited the multiplex and watched at least one of 3 shows. If 420 people watched A, 250 watched B, 450 watched C, what is the maximum number of people who watched all the 3 shows ?

(A) 0
(B) 50
(C) 60
(D) 100
(E) 250


Are You Up For the Challenge: 700 Level Questions
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Let the number of people who watched all the 3 shows = X and those who watched exactly 2 shows = Y

Formula: Total - Neither = A + B + C - (Exactly 2 categories) - 2(Exactly 3 categories)

—> 1000 - 0 = 450 + 250 + 420 - Y - 2X
—> 2X = 1120 - 1000 - Y
—> 2X = 120 - Y
—> X = 60 - Y/2

For X to be maximum, Y has to be minimum = 0

—> Maximum value if X = 60 - 0 = 60

IMO Option C

This is a fairly straightforward question on maximization of regions of a Venn diagram. In all maximization and minimization questions, remember that it’s not about taking the largest or the smallest value. It’s about taking the largest of the smallest value that does not make any region of the Venn diagram negative.

Let us go ahead and draw the Venn diagram now:



Since 1000 people visited the multiplex, the rectangle has 1000 people in total. The question also says that everyone watched at least one of the shows. This means everyone who visited the multiplex will be inside one of the circles. The area inside the rectangle but outside the circles will be ZERO.

In order to maximize the number of people who watched all 3, we will have to minimize the other regions. The minimum possible value of any region in a Venn diagram is 0.

From the diagram, we see that the sum of the sets = 420 + 250 + 450 = 1120. But, the sum of the sets contains the regions a, b and c twice and the region x thrice. If we remove a, b and c once and x twice from 1120, we get 1000 since this is the actual number of people who visited the multiplex (without repetition of any region). This means,
1120 –(a+b+c+2x) = 1000 which, on simplification, gives us,
a + b + c + 2x = 120. To maximize x, we take a+b+c = 0 and so, 2x = 120 or x = 60.

The correct answer option is C.
Hope this helps!
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Total= 1000
A= 420. B= 250. C= 450
Neither —0(zero)

The maximum number of people—Z(3 shows) —???

A= a+ a1+b1 +Z= 420
B= b+b1+ c1+Z= 250
C= c+c1+ a1+ Z= 450
——————————
a+b+c+a1+b1+c1+Z= 1000
a+b+c+ 2(a1+b1+c1)+ 3Z= 1120
—> (a1+b1+c1)+ 2Z= 120

—> if (a1+b1+c1)= 0, then 2Z= 120
—> Z= 60

The answer is C

Posted from my mobile device

answer should be d, we get (450+420+250)=1120 , now get maximum ppl who watched all show substitute value from given answer . it gets solved in 30 sec max .