Problem 72 - Precalculus: Mathematics for Calculus

Question

A small school has 100 students who occupy three classrooms: \(A, B,\) and \(C\). After the first period of the school day, half the students in room A move to room B, one-fifth of the students in room B move to room C and one-third of the students in room C move to room A. Nevertheless, the total number of students in each room is the same for both periods. How many students occupy each room?

Step-by-Step Solution

Answer

Rooms A, B, and C have 20, 50, and 30 students, respectively.

Step 1: Define Variables

Let the number of students in room A initially be \(x\), in room B be \(y\), and in room C be \(z\). Since the total number of students is 100, we have the equation:\[x + y + z = 100\]

Step 2: Set Up Movement Equations

According to the problem, after the movements between rooms, the number of students in each room doesn't change. Therefore, we setup the equations:1. In room A: \( x - \frac{1}{2}x + \frac{1}{3}z = x \).2. In room B: \( y + \frac{1}{2}x - \frac{1}{5}y = y \).3. In room C: \( z + \frac{1}{5}y - \frac{1}{3}z = z \).

Step 3: Simplify Equations

Simplify the equations from Step 2:1. For room A: \(\frac{1}{3}z = \frac{1}{2}x\).2. For room B: \(\frac{1}{2}x = \frac{1}{5}y\).3. For room C: \(\frac{1}{5}y = \frac{1}{3}z\).

Step 4: Solve System of Equations

Substitute from one simplified equation into another. Using the equation \(\frac{1}{3}z = \frac{1}{2}x\), substitute \(z = \frac{3}{2}x\) into the total equation:\[x + y + \frac{3}{2}x = 100\]Then solve:\[\frac{5}{2}x + y = 100\]Use \(\frac{1}{2}x = \frac{1}{5}y\) to get \(y = \frac{5}{2}x\).Substitute back into the equation:\[\frac{5}{2}x + \frac{5}{2}x = 100\]\[5x = 100\]\[x = 20\]Now find \(y\) and \(z\):\[y = \frac{5}{2} \times 20 = 50\]\[z = 100 - 20 - 50 = 30\]

Step 5: Verify Solution

Check that the solution satisfies all movement conditions:- From room A, \(20 - 10 + 10 = 20\).- From room B, \(50 + 10 - 10 = 50\).- From room C, \(30 + 10 - 10 = 30\).The values satisfy all conditions, confirming our solution.

Key Concepts

Algebra

Algebra is a powerful branch of mathematics that helps us express the world in numbers and symbols, allowing us to handle complex problems systematically. In this problem, using algebra means identifying unknown quantities, which we represent with variables. Here, we use variables like \(x\), \(y\), and \(z\) to denote the number of students in each classroom initially.

The core strength of algebra is its ability to model real-life situations with equations. By understanding the movements of students between the rooms, we can set up equations that reflect these changes. For example, the equation \(x + y + z = 100\) ensures that the total number of students remains constant. The usage of algebra not only helps establish these facts but also provides the tools needed to manipulate and solve them, thereby unveiling the desired outcomes.

Equation Solving

Solving equations is about finding the values of the unknowns that satisfy all given conditions. To solve a system of equations like the one in this problem, we perform several steps:

• We start with writing equations based on the movement constraints.
• Simplifying these into forms that can be more easily managed, such as changing \( \frac{1}{3}z = \frac{1}{2}x \) to \( z = \frac{3}{2}x \).
• Substitution is key. By substituting one equation into others, we narrow down the possibilities to find values that solve all equations simultaneously.

Once we solve these equations, we need to verify that these solutions hold true under all given conditions. This double-checking guarantees the solutions are accurate and that they adhere to the problem’s requirements.

Mathematical Modeling

Mathematical modeling involves translating real-world situations into mathematical language. This process is essential in problems like this one, where practical scenarios, such as the movement of students, are described mathematically.

Each classroom movement is modeled with equations, reflecting the exact numbers of students moving. With modeling:

• We define relationships between different entities (here, students in different classrooms).
• It helps visualize and simulate actions, allowing us to predict the outcomes of these actions before they occur.
• Through equations, a complicated scenario becomes an understandable, solvable task.

This approach not only aids in understanding the problem thoroughly but also encourages logical reasoning and critical thinking, crucial skills in both simple and complex problem-solving.