The problem provides data from a survey of 60 learners about their preferences for football, volleyball, and netball. The data includes the number of learners who liked each sport individually, the number who liked combinations of two sports, and the number who liked all three sports. The problem asks to illustrate the information on a Venn diagram and to answer the following questions: a) How many learners liked football? b) How many learners liked football and volleyball? c) How many learners liked exactly two games?

1. Problem Description

The problem provides data from a survey of 60 learners about their preferences for football, volleyball, and netball. The data includes the number of learners who liked each sport individually, the number who liked combinations of two sports, and the number who liked all three sports. The problem asks to illustrate the information on a Venn diagram and to answer the following questions:

a) How many learners liked football?

b) How many learners liked football and volleyball?

c) How many learners liked exactly two games?

2. Solution Steps

First, let's denote the sets of learners who liked football, volleyball, and netball as F, V, and N, respectively.

We are given the following information:

Total learners = 60

$|F| = 36$

$|V| = 38$

$|N| = 32$

$|F \cap V| = 21$

$|N \cap V| = 18$

$|F \cap N| = 12$

$|F \cap V \cap N| = 5$

a) How many learners liked football?

This is directly given in the data: 36 liked football. The Venn diagram seems to be incorrectly filled in the original image. According to the information in the picture $36 = 8 + 16 + 5 + 7$.

We are given that $|F| = 36$.

b) How many learners liked football and volleyball?

This is also directly given in the data: 21 liked football and volleyball. Thus, $|F \cap V| = 21$. The Venn diagram seems to represent $21-5 = 16$.

c) How many learners liked exactly two games?

To find the number of learners who liked exactly two games, we need to subtract the number of learners who liked all three games from the number of learners who liked each pair of games. Then we need to sum those values.

$|F \cap V| - |F \cap V \cap N| = 21 - 5 = 16$

$|V \cap N| - |F \cap V \cap N| = 18 - 5 = 13$

$|F \cap N| - |F \cap V \cap N| = 12 - 5 = 7$

The number of learners who liked exactly two games is the sum: $16 + 13 + 7 = 36$

3. Final Answer

a) Number of learners who liked football: 36

b) Number of learners who liked football and volleyball: 21

c) Number of learners who liked exactly two games: 36