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Chapter 2: Problem 86

Of 37 men and 33 women, 36 are teetotalers. Nine of the women are non-smokers and 18 of the men smoke but do not drink. 13 of the men and seven of the women drink but do not smoke. How many, at most, both drink and smoke.

Short Answer

Expert verified
At most, 12 people both drink and smoke.

Step by step solution

01

1. Understanding the given information

We have the following information: - Total number of people: 37 men + 33 women = 70 people - 36 teetotalers (people who don't drink) - 18 men smoke but don't drink (smokers and teetotalers) - 13 men drink but don't smoke - 9 women are non-smokers and also don't figure in the drinker's list (so, they are teetotalers) - 7 women drink but don't smoke
02

2. Organizing the information

Let's organize this information into sets: - A: Set of men who smoke but don't drink (18 people) - B: Set of men who drink but don't smoke (13 people) - C: Set of women who drink but don't smoke (7 people) - D: Set of women who neither smoke nor drink (9 people)
03

3. Finding the number of people in each category

From the given information, let's find the number of men and women who drink and/or smoke: - Number of men who drink or smoke: 37 (total men) - 18 (men who smoke but don't drink) - 13 (men who drink but don't smoke) = 6 - Number of women who drink or smoke: 33 (total women) - 9 (women who neither smoke nor drink) - 7 (women who drink but don't smoke) = 17
04

4. Using the principle of inclusion and exclusion

Now, let's find the maximum possible number of people who both smoke and drink using the principle of inclusion and exclusion. Total people who drink or smoke = \(|A| + |B| + |C|\) - number of people who both drink and smoke 36 (total teetotalers) = 6 + 17 - number of people who both drink and smoke number of people who both drink and smoke = 6 + 17 - 36 = -13 However, this number is negative and doesn't make sense. Thus, we must have made an error in our assumptions. Let's re-check the information.
05

5. Reassessing the information

Upon closer examination, we see that there is no specific information given about the women who drink and smoke (category A and C). As our aim is to find the maximum possible number of people who both smoke and drink, we can safely assume that all these 6 men who drink and smoke are both drinkers and smokers. This allows us to maximize the number of women who both drink and smoke.
06

6. Finding the maximum number of people who both drink and smoke

Now, we have: - 6 men who both drink and smoke - 10 women who smoke (33-23, since 23 do not smoke) - 6 women who drink (33-27, since 27 do not drink) The maximum possible number of women who both drink and smoke = min(10, 6) = 6 So, the maximum possible number of people who both drink and smoke = 6 (men) + 6 (women) = 12. Thus, at most, 12 people both drink and smoke.

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Most popular questions from this chapter

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