Question

A survey of 500 television viewers produced the following information:- 285 watch football, 195 watch hockey, 115 watch basketball, 45 watch football and basketball, 70 watch football and hockey, 50 watch hockey and basketball and 50 watch none of the three games. How many watch:
a. All the three games
b. Exactly one of the three games
c. Football and hockey, but not basketball
d. Football or hockey, but not basketball

Answer fast thw whole question

Answer 1

$\large\underline{\sf{Solution-}}$

It is given that,

A survey of 500 television viewers produced the following information:-

• 285 watch football,

• 195 watch hockey,

• 115 watch basketball,

• 45 watch football and basketball,

• 70 watch football and hockey,

• 50 watch hockey and basketball,

• 50 watch none of the three games

As,

50 watch none of three games and total viewers is 500.

So, Number of viewers watch atleast one of three games is 500 - 50 = 450.

Now,

Using Venn Diagram, [ See the attachment ]

We have,

$\rm :\longmapsto\:a + b + c + d + e + f + g = 450 - - (1)$

$\rm :\longmapsto\:a + d + e + g = 285 - - (2)$

$\rm :\longmapsto\:b + e + f + g = 195 - - (3)$

$\rm :\longmapsto\:c + d + f + g = 115 - - (4)$

$\rm :\longmapsto\:d + g = 45 - - (5)$

$\rm :\longmapsto\:e + g = 70 - - (6)$

.

$\rm :\longmapsto\:f + g = 50 - - (7)$

On Adding equation (2), (3) and (4), we get

$\rm :\longmapsto\:a + b + c + 2d + 2e + 2f + 3g = 595 - - (8)$

Now,

On Subtracting equation (1) from equation (8), we get

$\rm :\longmapsto\:d + e + f + 2g = 145 - - (9)$

On adding equation (5), (6) and (7), we get

$\rm :\longmapsto\:d + e + f + 3g = 165 - - (10)$

Now,

On Subtracting equation (9) from (10), we get

$\rm :\longmapsto\:g = 20$

Hence, on substituting g in equation (5), (6), (7), we get

$\rm :\longmapsto\:d = 25$

$\rm :\longmapsto\:e = 50$

$\rm :\longmapsto\:f = 30$

$\rm :\longmapsto\:a = 90$

$\rm :\longmapsto\:b = 95$

$\rm :\longmapsto\:c = 40$

Now,

(a) Number of viewers watch all three games = g = 20

(b) Number of viewers watch exactly one of the three games = a + b + c = 90 + 95 + 40 = 225

(c) Number of viewers watch Football and hockey, but not basketball = e = 50

(d) Number of viewers watch Football or hockey, but not basketball = a + e + b = 90 + 95 + 50 = 235