Question

A survey of 500 high school students was taken to determine their favorite chocolate candy. Of the 500 students surveyed, 149 like Snickers, 186 like Twix, 134 like Reese's Peanut Butter Cups, 70 like Snickers and Twix, 82 like Twix and Reese's Peanut Butter Cups, 31 like Snickers and Reese's Peanut Butter Cups, and 19 like all three kinds of chocolate candy. How many students like Reese's Peanut Butter Cups or Snickers, but not Twix?

Answer 1

Answer:

109 students like Reese's Peanut Butter Cups or Snickers, but not Twix.

Step-by-step explanation:

To solve this problem, we must build the Venn's Diagram of this set.

I am going to say that:

-The set A represents the students that like Snickers.

-The set B represents the students that like Twix.

-The set C represents the students that like Reese's Peanut Butter.

We have that:

$A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)$

In which a is the number of student that only like Snickers, $A \cap B$ is the number of students that like both Snickers and Twix, $A \cap C$ is the number of students that like both Reese's and Snickers. And $A \cap B \cap C$ is the number of students that like all these flavors.

By the same logic, we have:

$B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)$

$C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)$

How many students like Reese's Peanut Butter Cups or Snickers, but not Twix?

This are those who like any of these two or both. So:

$a + b + (A \cap C)$

We start finding the values from the intersection of three sets.

19 like all three kinds of chocolate candy. This means that

$(A \cap B \cap C) = 19$

31 like Snickers and Reese's Peanut Butter Cups: This means that

$(A \cap C) + (A \cap B \cap C) = 31$

$(A \cap C) = 12$

82 like Twix and Reese's Peanut Butter Cups

$(B \cap C) + (A \cap B \cap C) = 82$

$(B \cap C) = 73$

70 like Snickers and Twix

$(A \cap B) + (A \cap B \cap C) = 70$

$(A \cap B) = 51$

134 like Reese's Peanut Butter Cups

$C = 134$

$C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)$

$c + 12 + 73 + 19 = 134$

$c = 30$

149 like Snickers

$A = 149$

$A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)$

$a + 51 + 12 + 19 = 149$

$a = 67$

How many students like Reese's Peanut Butter Cups or Snickers, but not Twix?

$a + b + (A \cap C) = 67 + 30 + 12 = 109$

109 students like Reese's Peanut Butter Cups or Snickers, but not Twix.