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Misc 15 - In a survey of 60 people, 25 read newspaper H, 26 T

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Misc 15 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. Number of people who read newspaper H = n(H) = 25, Number of people who read newspaper T = n(T) = 26, Number of people who read newspaper I = n(I) = 26, Number of people who read both H & I = n(H ∩ I) = 9, Number of people who read both H & T = n(H ∩ T) = 11 Number of people who read both T & I = n(T ∩ I) = 8 Number of people who read all H ,T & I = n(H ∩ T ∩ I) = 3 Number of people who read at least one of the newspapers = n(H ∪ T ∪ I) We know that n(H ∪ T ∪ I) = n(H) + n(T) + n(I) – n(H ∩ T) – n(H ∩ I) – n(T ∩ I) + n(H ∩ T ∩ I) = 25 + 26 + 26 – 11 – 8 – 9 + 3 = 52 Hence, 52 people read at least one of the newspapers. Misc 15 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: (ii) The number of people who read exactly one newspaper. Let us draw a Venn diagram Let a denote the number of people who read newspapers H and T but not I. Let b denote the number of people who read newspapers I and H but not T Let c denote the number of people who read newspapers T and I but not H Let d denote the number of people who read all three newspapers. People who read exactly one news paper = n(H ∪ T ∪ I) – a – b – c – d d = n(H ∩ T ∩ I) = 3 n(H ∩ T) = a + d n(I ∩ T) = c + d n(H ∩ I) = b + d Adding the three equations n(H ∩ I) + n(I ∩ T) + n(H ∩ I) = 11 + 8 + 9 (a + d) + (c + d)+ (b + d) = 11 + 8 + 9 a + b + c + d + d + d= 28 a + b + c + d + 2d= 28 a + b + c + d = 28 – 2d a + b + c + d = 28 – 2 × 3 a + b + c + d = 28 – 6 a + b + c + d = 22 People who read exactly one news paper = n(H ∪ T ∪ I) – a – b – c – d = 52 – (a + b + c + d) = 52 – 22 = 30 Hence, 30 people read exactly one newspaper.

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.