Chapter 13.1, Problem 27AYU

Analyzing Survey Data In a consumer survey of $500$ people, $200$ indicated that they would be buying a major appliance within the next month, $150$ indicated that they would buy a car, and $25$ said that they would purchase both a major appliance and a car. How many will purchase neither? How many will purchase only a car?

To determine

How many will purchase neither, and how many will purchase only a car if in a consumer survey of $500$ people, $200$ indicated that they would be buying a major appliance within the next month, $150$ indicated that they would buy a car, and $25$ said that they would purchase both a major appliance and a car.

Answer to Problem 27AYU

Solution:

The number of people who will purchase neither is $175$ .

The number of people who will purchase only a car is $125$.

Explanation of Solution

Given information:

In a consumer survey of $500$ people, $200$ indicated that they would be buying a major appliance within the next month, $150$ indicated that they would buy a car, and $25$ said that they would purchase both a major appliance and a car.

Explanation:

Let $A=$ Set of consumers buying a major appliance.

$B=$ Set of consumers buying a car.

From given information,

$n\left(A\right)=200$, $n\left(B\right)=150$ and $n\left(A\cap B\right)=25$.

So $n\left(A\cap B\right)=25$ is the common part representing set $A$ and $B$.

The remaining portion of the circle representing set $A$ has $200-25=175$ elements.

The remaining portion of the circle representing set $B$ has $150-25=125$ elements.

Since $500$ people were surveyed, it follows that $500-175-25-125=175$ will purchase neither.

Use Venn diagram,

The number of people who will purchase neither is $175$.

The number of people who will purchase only a car is $125$.

Therefore, the number of people who will purchase neither is $175$ and the number of people who will purchase only a car is $125$.