There are 120 customers in a shop. 45 customers bought both bags and shoes. Every customer bought either bags or shoes. 11 more customers bought shoes than bags. (a) Illustrate this information in a Venn diagram. (b) Find the number of customers who bought shoes. (c) Calculate the probability that a customer selected at random bought bags.

1. Problem Description

There are 120 customers in a shop. 45 customers bought both bags and shoes. Every customer bought either bags or shoes. 11 more customers bought shoes than bags.

(a) Illustrate this information in a Venn diagram.

(b) Find the number of customers who bought shoes.

(c) Calculate the probability that a customer selected at random bought bags.

2. Solution Steps

(a) Venn Diagram:

Let $B$ be the set of customers who bought bags, and $S$ be the set of customers who bought shoes.

The total number of customers is 120, so $|B \cup S| = 120$.

The number of customers who bought both bags and shoes is 45, so $|B \cap S| = 45$.

(b) Number of customers who bought shoes:

Let $x$ be the number of customers who bought bags, so $|B| = x$.

Let $y$ be the number of customers who bought shoes, so $|S| = y$.

We are given that $y = x + 11$.

We know that:

$|B \cup S| = |B| + |S| - |B \cap S|$

$120 = x + y - 45$

$120 = x + (x + 11) - 45$

$120 = 2x + 11 - 45$

$120 = 2x - 34$

$2x = 120 + 34$

$2x = 154$

$x = \frac{154}{2} = 77$

So, the number of customers who bought bags is

7

7. Since $y = x + 11$, the number of customers who bought shoes is $y = 77 + 11 = 88$.

(c) Probability of buying bags:

The number of customers who bought bags is

7

7. The total number of customers is

1

2

0. The probability that a customer selected at random bought bags is $\frac{77}{120}$.

3. Final Answer

(a) Venn Diagram:

Two overlapping circles representing Bags (B) and Shoes (S). The overlapping region has

4

5. (b) The number of customers who bought shoes is

8

8. (c) The probability that a customer selected at random bought bags is $\frac{77}{120}$.