SOLUTION: In a survey of 290 newspaper readers, 181 of them read the Daily Times, 142 read the Guardian, 117 read Punch and each reads at least one of the three papers. If 75 read the Daily

Question 1206823: In a survey of 290 newspaper readers, 181 of them read the Daily Times, 142 read the Guardian, 117 read Punch and each reads at least one of the three papers. If 75 read the Daily Times and the Guardian, 60 read the Daily Times and Punch and 54 read the Guardian and Punch:
a) Draw a Venn diagram to illustrate this information.
b) How many readers read
i) all three papers,
ii) exactly two of the papers,
iii) exactly one of the papers,
iv) the Guardian alone?
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

Let x be the number who read all three newspapers. Then

75-x = number who read Daily Times and Guardian but not Punch
54-x = number who read Guardian and Punch but not Daily Times
60-x = number who read Punch and Daily Times but not Guardian

The number who read Daily Times only is (181) - ((75-x)+(60-x))-x = 46+x
The number who read Guardian only is (142) - ((75-x)+(54-x))-x = 13+x
The number who read Punch only is (117) - ((60-x)+54-x))-x = 3+x

Every one of the 290 readers reads at least one of the three:

(x)+(75-x)+(54-x)+(60-x)+(46+x)+(13+x)+(3+x)=290
251+x=290
x=39

You can draw the Venn diagram and use it to answer the questions.

Note that finding the number who read all three newspapers is found more easily using the inclusion-exclusion principle:

(sum of numbers who read at least one) - (sum of numbers who read two of the three) + (number who read all three) = total

(181+142+117)-(75+54+60)+x=290
440-189+x=290
261+x=290
x=39

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