Final answer:
To find the number of people who like both rock and country music in a survey of 200 people, we deducted the number of people who like neither and those who like only one genre from the total, which revealed that 55 people like both genres.
Step-by-step explanation:
The question asks us to determine how many people like both rock and country music given survey data from a sample of 200 people. We are provided with specific numbers: 35 people like only rock, 70 like only country, and 40 like neither. To find out the number of people who like both genres, we utilize the principle of inclusion and exclusion in set theory.
First, we'll calculate the total number of people who like at least one genre by subtracting the number of people who like neither from the total surveyed: 200 - 40 = 160. This number represents people who like rock, country, or both. Next, we add the number of people who like only rock and only country: 35 + 70 = 105. Now, we know that the number of people who like at least one genre (160) is the sum of those who like only rock, only country, and those who like both. So, to find the number of people who like both rock and country, we subtract the number of people who like only one genre from the total number of people who like at least one genre: 160 - 105 = 55. The answer is that 55 people like both rock and country music according to this survey data.