In a class, there are two sections $A$ and $B$. If 10 students of section $B$ shift over to section $A$, the strength of $A$ becomes three times the strength of $B$. But if 10 students shift over from $A$ to $B$, both $A$ and $B$ become equal in strength. How many students are there in sections $A$ and $B$?

## Concept & Logic This problem involves establishing a system of linear equations based on shifting quantities between two groups to find their initial values. Let the initial number of students in section $A$ be $a$ and in section $B$ be $b$. We can translate the given conditions into two algebraic equations. ## Step-by-Step Solution * **Condition 1:** If 10 students shift from $B$ to $A$: * Section $A$ becomes $a + 10$. * Section $B$ becomes $b - 10$. * The problem states: $a + 10 = 3(b - 10)$ $$ a + 10 = 3b - 30 $$ $$ a - 3b = -40 $$ (Equation 1) * **Condition 2:** If 10 students shift from $A$ to $B$: * Section $A$ becomes $a - 10$. * Section $B$ becomes $b + 10$. * The problem states they become equal: $a - 10 = b + 10$ $$ a - b = 20 $$ $$ a = b + 20 $$ (Equation 2) * **Solving the Equations:** * Substitute Equation 2 into Equation 1: $$ (b + 20) - 3b = -40 $$ $$ -2b + 20 = -40 $$ $$ -2b = -60 $$ $$ b = 30 $$ * Now, substitute the value of $b$ back into Equation 2: $$ a = 30 + 20 = 50 $$ ## Exam Strategy & Shortcut **Option Elimination** is by far the fastest method here! Simply test the options against the conditions: Take Option (d) 50, 30: Condition 2 check: If 10 shift from $A$ (50) to $B$ (30), $A$ becomes 40 and $B$ becomes 40. They are equal! Condition 1 check: If 10 shift from $B$ (30) to $A$ (50), $A$ becomes 60 and $B$ becomes 20. $60$ is exactly three times $20$. Both conditions match perfectly in under 10 seconds. ## Common Pitfall Students often mess up the signs when translating "shifting" into algebra. For example, writing $a + 10 = 3b$ instead of $a + 10 = 3(b - 10)$. You must remember that if one section *gains* students, the other section simultaneously *loses* that exact same number. ## Final Answer Therefore, the correct answer is **50, 30**.