A small school has 100 students who occupy three classrooms:...

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?

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Key Concepts

Fractional Movement

Fractional movement involves shifting a specific fraction of one group to another. Understanding and applying fractions correctly is crucial for determining how many individuals are transferred from one group to another, ensuring that the proportional changes are accurately incorporated in system equations.

Algebraic Manipulation

Algebraic manipulation includes techniques such as substitution, elimination, and simplification to solve systems of equations. This process is critical in step-by-step solving of balance problems, where fractions and total counts are integrated to find unknown quantities.

Conservation Principle

This principle states that in a closed system the total quantity remains constant even when distributed among different parts. In problems involving transfers or movements, such as the redistribution of students, the overall total does not change, which allows for the establishment of balance equations for each part.

System of Equations

A system of equations is used when multiple relationships between variables need to be satisfied simultaneously. In redistribution problems, equations represent the balance of students moving in and out of each classroom, and solving them together ensures that all conditions (like preserving total numbers) are met.

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