Question: Of the 84 farmers who attended a meeting at a Ruler Municipality, 35 volunteered to supervise the use of harmful pesticides in vegetable farming and 11 volunteered both in supervising and counselling not to use harmful pesticides. If the number of farmers who volunteered for counselling was 1.5 times the number of farmers who neither volunteered in supervision nor in counselling how many of the farmers volunteered for counselling?


Solution:

Let U be the set of total farmers who were surveyed.

Let S and C represent the sets of farmers who supervised the use of harmful pesticides in vegetable farming and counselling, respectively.

According to the question,
$n(U) = 84$
$n(S) = 35$
$n(C) = 1.5 × n(S \cup C)^c$
$n(S \cap U) = 11$

Now,
$n(U) = n(S) + n(C) - n(S \cap U) + n(S \cup C)^c$

$or, 84 = 35 + 1.5 × n(S \cup C)^c - 11 + n(S \cup C)^c$

$or, 84 = 24 + 2.5 × n(S \cup C)^c$

$or, 84 - 24 = 60 = 2.5 × n(S \cup C)^c$

$or, n(S \cup C)^c = \dfrac{60}{25}$

$\therefore n(S \cup C)^c = 24$


And,
$n(C) = 1.5 × n(S \cup C)^c$

$= 1.5 × 24$

$= 36$


Hence, the required number of farmers who were involved in counselling not to use harmful pesticides was 36.

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