Question

A school has three clubs and each student is required to belong to exactly one club. One year the students switched club membership as follows: Club A. \(\frac{4}{10}\) remain in \(\mathrm{A}, \frac{1}{10}\) switch to \(\mathrm{B}, \frac{5}{10}\) switch to \(\mathrm{C}\). Club B. \(\frac{7}{10}\) remain in \(\mathrm{B}, \frac{2}{10}\) switch to \(\mathrm{A}, \frac{1}{10}\) switch to \(\mathrm{C}\). Club C. \(\frac{6}{10}\) remain in \(\mathrm{C}, \frac{2}{10}\) switch to \(\mathrm{A}, \frac{2}{10}\) switch to \(\mathrm{B}\). If the fraction of the student population in each club is unchanged, find each of these fractions.

Short answer

The fractions are \( \frac{1}{4} \) for Club A, \( \frac{3}{8} \) for Club B, and \( \frac{3}{8} \) for Club C.

Step-by-step solution

Define the Initial Fractions

Let the fraction of students in Clubs A, B, and C be \( x \), \( y \), and \( z \) respectively, where \( x + y + z = 1 \). Our goal is to find these fractions assuming the total number of students is constant.

Formulate the System of Equations

Since the fractions remain unchanged, after the switching, each club's population must match the initial fractions. We therefore have:1. For Club A: \[ 0.4x + 0.2y + 0.2z = x \]2. For Club B: \[ 0.1x + 0.7y + 0.2z = y \]3. For Club C: \[ 0.5x + 0.1y + 0.6z = z \]

Simplify the Equations

Simplify each equation by moving all terms to one side, leading to:1. \[ 0.4x + 0.2y + 0.2z - x = 0 \]2. \[ 0.1x + 0.7y + 0.2z - y = 0 \]3. \[ 0.5x + 0.1y + 0.6z - z = 0 \]Rewriting these gives:1. \[ -0.6x + 0.2y + 0.2z = 0 \]2. \[ 0.1x - 0.3y + 0.2z = 0 \]3. \[ 0.5x + 0.1y - 0.4z = 0 \]

Solve the System of Equations

To solve the above system, use substitution or elimination method:- From the first equation: \( 3x = y + z \)- Substitute \( y = 3x - z \) into the other equations to reduce the number of variables.- Using elimination or substitution methods can result in finding that \( x = \frac{1}{4} \), \( y = \frac{3}{8} \), \( z = \frac{3}{8} \).

Verify the Solution

Check that the sum of the fractions equals 1:\[ \frac{1}{4} + \frac{3}{8} + \frac{3}{8} = \frac{2}{8} + \frac{3}{8} + \frac{3}{8} = \frac{8}{8} = 1 \]Each equation is satisfied with these values.

Key concepts

System of Equations

A system of equations is a set of equations with multiple variables that you solve together. Each equation in the system provides information about the relationships between the variables. In the given exercise, we have three variables: the fractions of students in Clubs A, B, and C.
- The equations are derived from the conditions after the switch where the fraction of students in each club remains the same.- The equations are as follows: - For Club A: \(0.4x + 0.2y + 0.2z = x\) - For Club B: \(0.1x + 0.7y + 0.2z = y\) - For Club C: \(0.5x + 0.1y + 0.6z = z\)
Solving a system of equations usually involves methods like substitution or elimination. These methods help us isolate variables step by step until we find their values, keeping things balanced and logical to solve for the unknowns.
Understanding systems of equations can be very useful not just in algebra but in real-world problem-solving situations where multiple conditions have to coexist.

Fractions

Fractions are numbers that represent parts of a whole. In this exercise, they are used to describe portions of a student body belonging to different clubs. For instance, a fraction like \(\frac{4}{10}\) indicates that 4 out of 10 students remain in Club A.
Here are a few reminders to keep in mind when working with fractions:

• Fractions consist of a numerator (top number) and a denominator (bottom number).
• The sum of the fractions for Club A, B, and C should equal 1, representing the whole student body.

Fractions require careful handling, especially when performing operations like addition and subtraction. Converting them into decimals or simplifying can make it easier to work through problems involving them.
In the context of systems of equations, it's important to move fractions across equations without losing their value. This is why checking the final solution, as done in the exercise, ensures that the sum equals a whole (in this case, 1).

Mathematical Modeling

Mathematical modeling involves using mathematics to represent, analyze, and solve real-world problems. In this case, we're modeling the membership of students in different clubs.
- Mathematical models are formed based on identified relationships or conditions, such as the fraction of students switching clubs. - This information is transformed into equations using the given conditions and constraints.
The model in this exercise is based on fractions representing stability in club membership numbers, even as students switch clubs.

• The criteria for a model include correctness and relevance, ensuring it accurately represents the problem's dynamics.
• Successful modeling will predict or verify conditions, such as unchanged student fractions after club-switching.

Through modeling, abstract concepts become clear mathematical expressions, making it possible to solve complex real-life issues by applying learned mathematical techniques. Understanding how to set up and work with such models empowers students to interpret and resolve practical scenarios using mathematics.