All the students of a school are made to stand in rows of 54 students each, and 30 such rows are formed. If instead the same students are arranged in rows of 45 students each, how many complete rows will be formed?

Introduction / Context:
This is a straightforward average and division style aptitude problem about arranging a fixed number of students into rows. We are first told how many rows are formed when there are 54 students in each row, and then asked how many rows would be formed if we change the row size to 45 students per row. The key idea is that the total number of students remains the same in both cases.

Given Data / Assumptions:

• Number of rows initially = 30.
• Number of students in each initial row = 54.
• Total number of students in the school is constant.
• We later arrange the same students in rows of 45 each.
• We must find the number of such 45-student rows.

Concept / Approach:
The core concept is that total students = number of rows * students per row. First, we calculate the total number of students using the initial arrangement. Then, with the second arrangement, we divide this total by 45 to obtain the new number of rows. Since the total number of students does not change, this method is both precise and efficient.

Step-by-Step Solution:
Step 1: Compute the total number of students using the first arrangement. Total students = 30 rows * 54 students per row. 30 * 54 = 30 * (50 + 4) = 1500 + 120 = 1620 students. Step 2: In the new arrangement, each row contains 45 students. Number of rows = total students / students per row = 1620 / 45. Step 3: Simplify 1620 / 45. Since 45 * 30 = 1350, the remaining students = 1620 - 1350 = 270. 45 * 6 = 270, so total rows = 30 + 6 = 36.
Verification / Alternative check:
Multiply back: 36 * 45 = (30 * 45) + (6 * 45) = 1350 + 270 = 1620 students. This matches the earlier total, so the calculation is consistent.
Why Other Options Are Wrong:
22, 26, 30 or 32 rows would correspond to 22 * 45, 26 * 45, 30 * 45 or 32 * 45 students, none of which equals 1620, so they cannot represent the same total group.
Common Pitfalls:
Sometimes students try to directly proportion 54 and 45 without first computing the total, which can lead to mistakes. Another error is rounding instead of performing exact integer division, even though the question clearly expects a whole number of rows.
Final Answer:
The number of rows formed when students stand 45 in a row is 36.