Theory question 108
Sets(a) In a market survey, 100 traders sell fruits. 40 sell apples, 46 oranges, 50 mangoes, 14 apples and oranges, 15 apples and mangoes and 10 sell the three fruits. Each of the 100 traders sells at least one of the three fruits. (i) Represent the information in a Venn diagram. (ii) Find the number that sell oranges and mangoes only.
(b) Find the value of x for which 312four + 52x = 96ten.
Show the model answer
(a)(i) Let y sell oranges and mangoes only. Regions: all three 10; apples and oranges only 4; apples and mangoes only 5; apples only 40 − 19 = 21; oranges only 46 − 14 − y = 32 − y; mangoes only 50 − 15 − y = 35 − y.
(ii) Total: 21 + 4 + 5 + 10 + y + (32 − y) + (35 − y) = 100, so 107 − y = 100 and y = 7.
(b) 312four = 48 + 4 + 2 = 54, so 52x = 42: 5x + 2 = 42 and x = 8.
Fill the Venn diagram from the centre (all three) outwards.