A child has 1,200 five rupee coins and wants to arrange them in a rectangular pattern such that the number of rows equals the number of columns, that is, in a perfect square array. What is the minimum number of additional coins he needs so that this arrangement becomes possible?

Introduction / Context:
This arithmetic reasoning question is about perfect squares. The child has 1,200 identical coins and wants to arrange them in a square array such that the number of rows equals the number of columns. Because 1,200 is not a perfect square, he must add some more coins. You are asked to find the smallest number of additional coins required to make the total coin count a perfect square.

Given Data / Assumptions:

• The child initially has 1,200 coins.
• An arrangement in which rows equal columns requires a perfect square number of coins.
• We seek the least non negative integer x such that 1,200 + x is a perfect square.
• The answer must be chosen from the given options.

Concept / Approach:
The idea is to find the smallest perfect square greater than 1,200. The additional coins needed will be the difference between that perfect square and 1,200. To do this efficiently, compute the integer square root of 1,200, find the next integer, and square it. The result is the next perfect square after 1,200. This is a standard trick in such questions and avoids listing squares one by one.

Step-by-Step Solution:
Step 1: Find the largest integer whose square is less than or equal to 1,200. Step 2: 34^2 = 1,156 and 35^2 = 1,225. So 34^2 is less than 1,200 and 35^2 is greater than 1,200. Step 3: Therefore, the next perfect square after 1,200 is 35^2 = 1,225. Step 4: The number of additional coins required is 1,225 - 1,200. Step 5: Compute the difference: 1,225 - 1,200 = 25. Step 6: Hence, the child must add 25 more coins to have a total that is a perfect square.

Verification / Alternative check:
Check that 1,225 coins can indeed be arranged as a 35 by 35 square, because 35 rows of 35 coins each gives exactly 35 * 35 = 1,225 coins. Also verify that 1,200 itself is not a perfect square by noting that 34^2 = 1,156 and 35^2 = 1,225, so there is no integer whose square is 1,200. Thus, 25 is the smallest addition required.

Why Other Options Are Wrong:
Option A 125 would take the total to 1,325, which is not a perfect square. Option B 96 would give a total of 1,296, which is 36^2, but it is more than the minimum 25 required to reach the smaller perfect square 1,225. Option C 35 raises the total to 1,235, which is not a perfect square. The problem asks for the minimum, so 25 is the only correct answer.

Common Pitfalls:
Some students do not look for the nearest perfect square and instead jump to a larger square like 36^2, thereby adding more coins than necessary. Others may confuse the process and try to subtract coins to reach a lower perfect square, which is not allowed in this problem. Remember that you must add the smallest amount to reach the next perfect square above the current total.

Final Answer:
The minimum number of additional coins needed is 25.