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There are 500 children in a village. 49% of children watch Doraemon and 53% of children watch Sinchain and 62% of children watch Tom & Jerry. 27% of children watch both Doraemon and Sinchain. 29% of children watch both Sinchain and Tom & Jerry. 28% of children watch both Doraemon and Tom & Jerry. 5% of children don’t watch any cartoons. Find the number of children who watch at least two of the given cartoons.

There are 500 children in a village. 49% of children watch Doraemon and 53% of children watch Sinchain and 62% of children watch Tom & Jerry. 27% of children watch both Doraemon and Sinchain. 29% of children watch both Sinchain and Tom & Jerry. 28% of children watch both Doraemon and Tom & Jerry. 5% of children don’t watch any cartoons. Find the number of children who watch at least two of the given cartoons. Correct Answer 270

Formula used:

Total number of children who watch all cartoons

⇒ n(A ∪ B ∪ C) = n(A) + n (B) + n (C) - n (A ∩ B) - n (B ∩ C) - n (C ∩ A) + n (A ∩ B ∩ C)

Where n(A) = number of elements in set A.

Watch at least two cartoons = n(A ∩ B) + n(B ∩ C) + n(C ∩ A) - 2 × n(A ∩ B ∩ C)

Given:

Total number of children in village = 500

Who watch Doraemon = 49%

Who watches Sinchain = 53%

Who watch Tom & Jerry = 62%

Who watches both Doraemon and Sinchain = 27%

Who watches both Sinchain and Tom & Jerry = 29%

Who watches both Doraemon and Tom & Jerry = 28%

Who don’t watch any cartoons = 5%

Calculation:

A

Doraemon

B

Sinchain

C

Tom & Jerry

 

Total number of children in village = 500

Total number of children who watch Doraemon = 49% of 500

⇒ (49 / 100) × 500 = 245

Total number of children who watch Sinchain = 53% of 500

⇒ (53 / 100) × 500 = 265

Total number of children who watch Tom & Jerry = 62% of 500

⇒ (62 / 100) × 500 = 310

Total number of children who watch both Doraemon and Sinchain = 27% of 500

⇒ (27 / 100) × 500 = 135

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Total number of children who watch both Sinchain and Tom & Jerry = 29% of 500

⇒ (29 / 100) × 500 = 145

Total number of children who watch both Doraemon and Tom & Jerry = 28% of 500

⇒ (28 / 100) × 500 = 140

Total number of children who don’t watch any cartoons = 5% of 500

⇒ (5 / 100) × 500 = 25

Total children who watch cartoons = 500 - 25 = 475

Total number of children who watch all cartoons

By formula,

⇒ n(A ∪ B ∪ C) = n(A) + n (B) + n (C) - n(A ∩ B) - n(B ∩ C) - n(C ∩ A) + n(A ∩ B ∩ C)

⇒ 475 = 245 + 265 + 310 - 135 - 145 - 140 + n (A ∩ B ∩ C)

⇒ n(A ∩ B ∩ C) = 475 - 400 = 75

Now number of children who watch at least two cartoons

⇒ n(A ∩ B) + n(B ∩ C) + n(C ∩ A) - 2 × n(A ∩ B ∩ C)

⇒ 135 + 145 + 140 - 2 × 75

⇒ 420 - 150 = 270

∴ The total number of children who watch at least two cartoons is 270.

Alternate method:

By percentage:

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Let the total of children in the village be 100%

Children who watch Doraemon = 49%

Children who watch Sinchain = 53%

Children who watch Tom & Jerry = 62%

Since 5% of children don’t watch any cartoons.

Total children who watch cartoons = 100 - 5 = 95%

Who watch both Doraemon and Sinchan = 27%

Who watch both Sinchan and Tom & Jerry = 29%

Who watch both Doraemon and Tom & Jerry = 28%

Total number of children who watch all cartoons

⇒ 95 - 49 - 53 - 62 + 27 + 29 + 28 = 15%

Now number of children who watch at least two cartoons

⇒ 27 + 29 + 28 - 2 × 15 = 54%

⇒ 54% of 500

⇒ (54 / 100) × 500 = 270

∴ The total number of children who watch at least two cartoons is 270.

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