Question

A survey shows that 63% people watch news channel A whereas 76% people watch news channel B. If x% of people watch both news channels, then prove that $ 39 \leqslant x \leqslant 63 $

Answer 1

Hint: 63% people watch news channel ‘A’ and 76% people watch news channel ‘B’. This means 63 people out of 100 watches channel A and 76 people out of 100 watches channel B. This can also be written as $ n\left( A \right) = 63 $ and $ n\left( B \right) = 76 $ . So the number of people who watch both the channels is $ A \cap B $ . So first write the formula of $ n\left( {A \cup B} \right) $ and find the value of $ n\left( {A \cap B} \right) $ (percentage of people who watch both the channels.
Formula used:
 $ n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right) $

Complete step-by-step answer:
We are given that a survey shows that 63% people watch news channel A whereas 76% people watch news channel B. We have to prove that $ 39 \leqslant x \leqslant 63 $ where x is the percentage of people who watch both news channels.
No. of people who watch channel A out of 100 is 63 and the no. of people who watch channel B out of 100 is 76.
This means $ n\left( A \right) = 63 $ and $ n\left( B \right) = 76 $
No. of people who watches both the channels is $ A \cap B $ and let x be $ n\left( {A \cap B} \right) $
We already know that $ n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right) $
Substituting the know values in the above formula, we get $ n\left( {A \cup B} \right) = 63 + 76 - n\left( {A \cap B} \right) $
But we know that the total no. of people cannot exceed 100 while taking percentage.
This means $ n\left( {A \cup B} \right) \leqslant 100 $
So the formula becomes $ \left[ {63 + 76 - n\left( {A \cap B} \right)} \right] \leqslant 100 $
 $ \Rightarrow - n\left( {A \cap B} \right) \leqslant 100 - 139 $
 $ \Rightarrow - n\left( {A \cap B} \right) \leqslant - 39 $
 $ \Rightarrow n\left( {A \cap B} \right) \geqslant 39 $
We have considered $ n\left( {A \cap B} \right) $ as x
Therefore, $ x \geqslant 39 $
But we already know that $ n\left( {A \cap B} \right) $ will always be less than or equal to $ n\left( A \right) $ and $ n\left( B \right) $
This means $ \Rightarrow n\left( A \right) \geqslant n\left( {A \cap B} \right) $
 $ \Rightarrow n\left( A \right) \geqslant x $
 $ \Rightarrow n\left( A \right) \geqslant x \geqslant 39 $
We know the value of $ n\left( A \right) $ is 63.
 $ \therefore 63 \geqslant x \geqslant 39 \Leftrightarrow 39 \leqslant x \leqslant 63 $
Hence, proved.

Note: Always remember that when an inequality is multiplied by a negative sign, greater than sign becomes less than and less than sign becomes greater than, which means the direction of the inequality changes.