When all the students in a school are made to stand in rows of 54, 30 such rows are formed. If the students are made to stand in rows of 45, how many such rows will be formed?

### Concept & Strategy This is an inverse proportion problem involving area/element conservation. The total number of students in the school is constant regardless of how they are arranged. $$\text{Total Students} = (\text{Students per Row}) \times (\text{Number of Rows})$$ ### Step-by-Step Solution * **Given:** Arrangement 1 has 54 students per row and forms 30 rows. Arrangement 2 has 45 students per row. Let the required number of rows be $R$. * First, find the total number of students in the school using the first arrangement: $$\text{Total Students} = 54 \times 30$$ $$\text{Total Students} = 1620$$ * Set up the equation for the second arrangement using the same total: $$45 \times R = 1620$$ * Solve for $R$: $$R = \frac{1620}{45}$$ * Simplify the division: $$R = 36$$ ### Exam Strategy & Shortcut Avoid calculating the massive total ($54 \times 30 = 1620$). Instead, set it up as a single fraction and simplify using common factors. $R = \frac{54 \times 30}{45}$ Both 30 and 45 are divisible by 15. $R = \frac{54 \times 2}{3}$ Now divide 54 by 3. $R = 18 \times 2 = 36$. This keeps the math strictly in the realm of basic mental multiplication tables. ### Common Pitfall Attempting to solve via direct proportion (e.g., as row size goes down, row count goes down). This is an *inverse* relationship: making the rows smaller means you will need *more* rows to fit everyone. ### Final Answer Therefore, the correct answer is **36**.