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

Introduction / Context:
This problem uses the basic concept of average and multiplication to count how many students there are in total and how rearranging them into different sized rows changes the number of rows. It is a typical counting and grouping puzzle from school level aptitude.

Given Data / Assumptions:
- When students stand in rows of 54, there are 30 such rows.
- The total number of students remains the same for any arrangement.
- In the second arrangement, each row contains 45 students.
- We need to find how many rows of 45 students can be formed.

Concept / Approach:
The total number of students is equal to number of rows multiplied by students per row. First we find the total number of students using the first arrangement. Then, for the second arrangement, we divide the same total by 45 to find the new number of rows. There is no rounding because the problem is designed so that the total divides exactly.

Step-by-Step Solution:
Step 1: Compute the total number of students from the first arrangement: total = 54 * 30. Step 2: Calculate 54 * 30 = 1620 students. Step 3: In the second arrangement, each row has 45 students. Step 4: The number of rows formed is total students divided by students per row, that is 1620 / 45. Step 5: Compute 1620 / 45 = 36 rows.

Verification / Alternative check:
You can quickly check by multiplying 36 * 45. This gives 1620, the same total number of students as before. Since the total is unchanged, 36 rows of 45 students each is consistent with the original arrangement.

Why Other Options Are Wrong:
- Option 22: 22 * 45 = 990, which is less than the total 1620, so many students would be left out.
- Option 32: 32 * 45 = 1440, still less than 1620, so it cannot be the correct number of rows.
- Option 26: 26 * 45 = 1170, which again does not match the required total.

Common Pitfalls:
Some students confuse the two numbers 54 and 30 and mistakenly divide instead of multiplying to find the total. Another error is to try to adjust the number of rows directly without first working out the total. Always remember that when the group size changes but the total remains constant, multiply to find the total first and then divide by the new group size.

Final Answer:
The students will form 36 rows when arranged with 45 students in each row.