Question

In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T, 26 read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find:
(i) the number of people who read at least one of the newspapers.
(ii) the number of people who read exactly one newspaper.

Answer 1

Refer to the Venn diagram given below.

n(U) = a+b+c+d+e+f+g+h = 60 ...(i)
n(H) = a+b+c+d = 25 ...(ii)
n(T) = b+c+f+g = 26 ...(iii)
n(I) = c+d+e+f = 26 ...(iv)
$n(H\cap I)=c+d=9$ ...(v)
$n(H\cap T)=b+c=11$ ...(vi)
$n(T\cap I)=c+f=8$ ...(vii)
$n(H\cap T\cap I)=c=3$ ...(viii)



Putting value of c in (vii),
$3+f=8\Rightarrow f=5$
Putting value of c in (vi),
$3+b=11\Rightarrow b=8$
Putting value of C in (v)
$3+d=9\Rightarrow d=6$
Putting value of c, d, f in (iv),
$3+6+e+5=26\Rightarrow e=26-14=12$
Putting value of b, c, f in (iii),
$8+3+5+g=26\Rightarrow g=26-16=10$
Putting value of b, c, d in (ii),
$a+8+3+6=25\Rightarrow a=25-17=8$
(i) Number of people who read at least one of the three newspapers
=a+b+c+d+e+f+g
=8+8+3+6+12+5+10
=52

(ii) Number of people who read exactlyone newspapers
=a+e+g
=8+12+10=30