In a particular locality, 25 people like walking, 22 people like running, and 15 people like cycling. 11 people like both walking and running, 9 people like both running and cycling, and 7 people like both walking and cycling. 3 people like all three activities, while 1 do not like any of the activities. We need to: (a) Draw a Venn diagram to illustrate this information. (b) Find how many people are in this locality. (c) Find how many people like walking only. (d) Find how many people like two different activities only.

1. Problem Description

In a particular locality, 25 people like walking, 22 people like running, and 15 people like cycling. 11 people like both walking and running, 9 people like both running and cycling, and 7 people like both walking and cycling. 3 people like all three activities, while 1 do not like any of the activities. We need to:

(a) Draw a Venn diagram to illustrate this information.

(b) Find how many people are in this locality.

(c) Find how many people like walking only.

(d) Find how many people like two different activities only.

2. Solution Steps

(a) Venn Diagram:

Let W be the set of people who like walking, R be the set of people who like running, and C be the set of people who like cycling. We fill in the Venn diagram from the inside out.

* $W \cap R \cap C = 3$

* $W \cap R = 11$, so $W \cap R \cap C' = 11 - 3 = 8$

* $R \cap C = 9$, so $R \cap C \cap W' = 9 - 3 = 6$

* $W \cap C = 7$, so $W \cap C \cap R' = 7 - 3 = 4$

* $W = 25$, so $W \text{ only } = 25 - 8 - 3 - 4 = 10$

* $R = 22$, so $R \text{ only } = 22 - 8 - 3 - 6 = 5$

* $C = 15$, so $C \text{ only } = 15 - 4 - 3 - 6 = 2$

* Number of people who do not like any activity = 1

(b) Total number of people in the locality:

$|W \cup R \cup C|$ = (W only) + (R only) + (C only) + (W and R only) + (R and C only) + (W and C only) + (All three) + (None)

$= 10 + 5 + 2 + 8 + 6 + 4 + 3 + 1 = 39$

(c) Number of people who like walking only:

From the Venn diagram, the number of people who like walking only is $10$.

(d) Number of people who like two different activities only:

This is the sum of the number of people who like walking and running only, running and cycling only, and walking and cycling only.

$= 8 + 6 + 4 = 18$

3. Final Answer

(a) Venn Diagram: (See explanation above for the values in each region.)

(b) Total number of people in the locality: 39

(c) Number of people who like walking only: 10

(d) Number of people who like two different activities only: 18