Question

A survey was conducted by a website on which online music channels subscribers use on a regular basis. The following information summarizes the answers. 11 listened to rap, heavy metal, and alternative rock. 15 listened to rap and heavy metal. 30 listened to heavy metal and alternative rock. 19 listened to rap and alternative rock. 26 listened to rap. 48 listened to heavy metal. 55 listened to alternative rock. 22 listened to none of these three channels. a. How many people were surveyed? b. How many people listened to either rap or alternative rock? c. How many listened to heavy metal only?

Answer 1

Let's use a Venn diagram to solve this problem. We have three circles representing the three music channels: rap, heavy metal, and alternative rock. The intersection of all three circles represents the number of people who listen to all three channels, which is 11. The intersection of the rap and heavy metal circles represents the number of people who listen to both rap and heavy metal, but not alternative rock, which is 15 - 11 = 4. Similarly, the intersection of the heavy metal and alternative rock circles represents the number of people who listen to both heavy metal and alternative rock, but not rap, which is 30 - 11 = 19. The intersection of the rap and alternative rock circles represents the number of people who listen to both rap and alternative rock, but not heavy metal, which is 19 - 11 = 8.

a. The total number of people surveyed is equal to the sum of the number of people in each circle plus the number of people who listen to none of these three channels: 26 + 48 + 55 - (4 + 19 + 8 + 2 * 11) + 22 = 118.

b. The number of people who listened to either rap or alternative rock is equal to the sum of the number of people in the rap circle plus the number of people in the alternative rock circle minus the intersection of these two circles: 26 + 55 - (8 + 11) = 62.

c. The number of people who listened to heavy metal only is equal to the total number of people in the heavy metal circle minus the intersections with the other two circles: 48 - (4 + 19 + 11) = 14.