Question

A coffee shop collected the following information regarding purchases from 110 of its customers 59 purchased coffee 39 purchased donuts 16 purchased coffee and donuts. Complete parts a) through c) Of those surveyed, how many purchased only coffee? Of those surveyed, how many purchased only donuts? Of those surveyed, how many did not purchase either of these items?

Answer 1

Answer:

Let X be the number of customer purchased coffee

Let Y be the number of customer purchased donuts

Then $X\cap Y$ is the number of customer purchased both coffee and donuts

$X\cup Y$  is the number of customer purchased both coffee or donuts

The Number of customers purchased  only Coffee:

Number of customers purchased  only donuts =n(X -Y)

$n( X -Y) = n(X) - n(X \cap Y)$

n(X-Y) = 59 -16

n(Y - X) = 43

The Number of customers purchased  only donuts:

Number of customers purchased  only donuts =n(Y -X)

$n(Y - X) = n(Y) - n(X \cap Y)$

n(Y - X) = 39 -16

n(Y - X) = 23

The Number of customers did not purchase either of these items:

Number of customers did not purchase either of these items = $n(X\cup Y)^{\prime}$

First lets find   $n(X\cup Y)$

$n(X\cup Y) = n(X) + n(Y) - n( X \cap Y)$

$n(X\cup Y) = 59 + 39 - 16$

$n(X\cup Y) = 82$

$n(X\cup Y)^{\prime}$  =$110 - n(X\cup Y)$  = 28