There are benches in a classroom. If 4 students sit on each bench, then three benches are left vacant and if 3 students sit on each bench, then 3 students are left standing. The total number of students in the class is

### Concept & Logic Similar to the previous problem, establish two linear equations representing the total number of students to find the unknown number of benches. ### Step-by-Step Solution * **Given:** Let the total number of benches in the classroom be $x$. * **Condition 1:** If 4 students sit per bench, 3 benches are left vacant. This means the students are occupying exactly $(x - 3)$ benches. $$\text{Total Students} = 4(x - 3)$$ * **Condition 2:** If 3 students sit per bench, 3 students are left standing. This means 3 students don't have a seat after all $x$ benches are filled. $$\text{Total Students} = 3x + 3$$ * **Equate the two expressions:** $$4(x - 3) = 3x + 3$$ $$4x - 12 = 3x + 3$$ * **Solve for $x$:** $$4x - 3x = 3 + 12$$ $$x = 15 \text{ (Total benches)}$$ * **Find total students:** Substitute $x = 15$ into the second equation: $$\text{Total Students} = 3(15) + 3 = 45 + 3 = 48$$ ### Exam Strategy & Shortcut Use option verification. The total number of students minus 3 must be perfectly divisible by 3 (from condition 2), and the total number of students must be perfectly divisible by 4 (since 4 sit on each occupied bench). Check the options: (a) 36: Divisible by 4. $36 - 3 = 33$ (divisible by 3). Benches = 9 or 11. Conflict. (b) 42: Not perfectly divisible by 4. (c) 48: $48 / 4 = 12$ occupied benches. Total benches = $12 + 3 = 15$. Now check condition 2: $15 \times 3 = 45$ seated + 3 standing = 48. Perfect match. ### Common Pitfall Setting up the first equation incorrectly as $4x - 3$ instead of grouping the vacant benches as $4(x - 3)$. The number 4 multiplies the *number of occupied benches*, not the total benches minus 3 individuals. ### Final Answer Therefore, the correct answer is **48**.