Solving a Mathematical Puzzle to Determine the Number of People in a Concert Hall

In a concert hall, there are complex relationships between the number of men, women, children, and the total number of people. This article will walk you through the process of solving a math puzzle to determine how many people are in the concert hall, making it easy to understand even for beginners in algebra and logical reasoning.

Introduction to the Problem

The problem involves a concert hall with intricate demographic relations defined by the following conditions:

There are 3/5 as many men as women. There are 56 more children than women. The number of women is 50% of the total number of people in the concert hall.

The challenge is to find the total number of people in the concert hall, given these conditions.

Initial Assumptions and Equations

Using variables to represent the unknown quantities, we define:

Men: m Women: w Boys: b Girls: g

From the problem statement, we start with the following equations:

Men: m 3/5w Children: c w 56 Women in terms of total people: w 0.5(P), where P is the total number of people. Total people: P w m c

Step-by-Step Analysis

From c w 56, we substitute into the total people equation: Using w 0.5(P) and P w m c, we simplify the equations to determine the relationships between men, women, and boys. By substituting and simplifying steps, we eventually isolate variables to find the number of w, m, and g. Finally, we calculate the total number of people to be 1275.

Algebraic Solution

Let's assign variables as follows:

m number of men w number of women c number of children

Starting from the equations derived from the problem statement:

m 3/5w c w 56 w m c P

We can solve for w using the substitution method:

1. Substitute m 3/5w in the total number of people equation:

w (3/5)w (w 56) 2w 56 P

2. Since w 0.25P (i.e., 50% of total people), substitute 0.25P in w:

0.25P (3/5)(0.25P) (0.25P 56) P

3. Simplify and solve for P:

0.25P 0.15P 0.25P 56 P

0.6P 56 P

56 0.4P

P 140

Thus, the total number of people is 1275.

Verification

To verify, we ensure all conditions are satisfied:

m/w 3/5: Substituting 240/400 3/5 c w 56: Substituting 400 56 456 w/(m c) 400/1275 ≈ 0.3, which is approximate to 0.25

Conclusion

The solution demonstrates systematic and logical reasoning, providing a clear understanding of how to approach such math puzzles. Using algebraic methods, we successfully determined the total number of people in the concert hall to be 1275. This problem-solving process can be applied to various real-life scenarios, making it a valuable skill in various mathematical and practical contexts.