(2) $\frac{3}{5}$ of the men at a club meeting drank beer and $\frac{5}{8}$ drank wine. Every man drank at least one of these drinks. If 18 men drank both beer and wine, how many men attended the club meeting?
Answer (1)
Let x be the total number of men.
Express the number of men who drank beer and wine in terms of x : 5 3 x and 8 5 x respectively.
Use the principle of inclusion-exclusion to set up the equation: x = 5 3 x + 8 5 x − 18 .
Solve for x to find the total number of men: 80 .
Explanation
Understanding the Problem Let's analyze the problem. We are given that 5 3 of the men drank beer, 8 5 drank wine, and 18 men drank both. We need to find the total number of men at the meeting, knowing that everyone drank at least one of the beverages.
Defining Variables and Sets Let x be the total number of men at the club meeting. Let B be the number of men who drank beer and W be the number of men who drank wine. We are given:
B = 5 3 x W = 8 5 x B ∩ W = 18
Since every man drank at least one of the drinks, the total number of men is the number who drank beer or wine, which is B ∪ W = x .
Applying Inclusion-Exclusion Using the principle of inclusion-exclusion, we have:
∣ B ∪ W ∣ = ∣ B ∣ + ∣ W ∣ − ∣ B ∩ W ∣
Substituting the given values, we get:
x = 5 3 x + 8 5 x − 18
Solving for x Now, we solve for x :
x = 5 3 x + 8 5 x − 18 x = 40 24 x + 40 25 x − 18 x = 40 49 x − 18 18 = 40 49 x − x 18 = 40 49 x − 40 40 x 18 = 40 9 x
Calculating the Total Number of Men Multiplying both sides by 9 40 , we get:
x = 18 × 9 40 x = 2 × 40 x = 80
Final Answer Therefore, there were 80 men at the club meeting.
Examples
Imagine you're organizing a party and need to figure out how many people are coming based on RSVPs for different food options. If 3/5 of the attendees want pizza, 5/8 want burgers, and 18 want both, you can use the same math to determine the total number of guests. This helps you plan the right amount of food and drinks, ensuring everyone has a great time. Understanding set theory and inclusion-exclusion principle is useful in event planning, resource allocation, and data analysis to avoid double-counting and make accurate estimations.
