Question

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

At most, there are 24 people who both drink and smoke in the group.

Step-by-step solution

Define the categories and create a two-way table

We can categorize people into four groups: 1. Men who smoke (M_S) 2. Men who drink (M_D) 3. Women who smoke (W_S) 4. Women who drink (W_D) We will create a two-way table to represent these groups and their intersections (i.e. smoking and drinking): | | Men Smoke | Men Drink | Total Men | |-----------|-----------|-----------|-----------| | Women Smoke | | | | | Women Drink | | | | | Total Women | | | |

Fill in the given information in the table

Fill in the table with given data: - 18 men smoke but do not drink (M_S∩M_D') - 13 men drink but do not smoke (M_S'∩M_D) - 9 women are non-smokers (W_S') - 7 women drink but do not smoke (W_S'∩W_D) | | Men Smoke | Men Drink | Total Men | |-----------|-----------|-----------|-----------| | Women Smoke | | | | | Women Drink | | 7 | | | Total Women | | 13 | 33 | |-----------|-----------|-----------|-----------| | Total Men | 18 | | 37 | The table is partially filled now.

Calculate the remaining values in the table

Now, we will use the Principle of Inclusion-Exclusion (PIE) to fill the remaining values in the table. Calculate Total Men who smoke (M_S): Men who smoke but do not drink = 18 Therefore, M_S = 18. Calculate Total Men who drink (M_D): M_D = Total Men who drink - Men who drink but do not smoke M_D = 37 - 13 M_D = 24 Calculate Total Women who smoke (W_S): W_S = Total Women - Women who are non-smokers W_S = 33 - 9 W_S = 24 Calculate Total Women who drink (W_D): W_D = Total Women who drink - Women who drink but do not smoke W_D = 33 - 7 W_D = 26 Update the table: | | Men Smoke | Men Drink | Total Men | |-----------|-----------|-----------|-----------| | Women Smoke | | | | | Women Drink | | 7 | | | Total Women | 24 | 26 | 33 | |-----------|-----------|-----------|-----------| | Total Men | 18 | 24 | 37 | Now there is only one cell left which is for people who both drink and smoke.

Calculate the maximum number of people who both drink and smoke

According to the problem, there are a total of 36 teetotalers in the group. Based on what we have found so far, sum of non smoking men and women = 18 + 9 = 27. Let x be the number of people who both drink and smoke: Total teetotalers = Total men + Total women - People who drink and smoke - People who neither drink nor smoke 36 = 37 + 33 - x - 27 x = 24 Now, our table is complete: | | Men Smoke | Men Drink | Total Men | |-----------|-----------|-----------|-----------| | Women Smoke | | | | | Women Drink | | 7 | | | Total Women | 24 | 26 | 33 | |-----------|-----------|-----------|-----------| | Total Men | 18 | 24 | 37 | |-----------|-----------|-----------|-----------| | Both | | | 24 | So, at most, there are 24 people who both drink and smoke in the group.