Question

An army officer wishes to arrange $4770$ soldiers in the form of a square. After arranging he finds that $9$ soldiers are left out. Find the number of soldiers in each row.

Answer 1

Hint: The total number of soldiers given is $4770$ , but the number of soldiers in the square is $4761$ . We assume the number of soldiers in each row to be “x”. Now, the number of soldiers in the square will be ${{x}^{2}}=4761$ , which can be solved to get the answer.

Complete step-by-step solution:
In this problem, we are given that the total number of soldiers is $4770$ . It is also said that the soldiers have been arranged in the form of a square. After doing so, the number of soldiers that are left out is $9$ . Now, let the number of rows be “a” and the number of columns be “b”. But, since the arrangement is a square, the number of rows must be equal to the number of columns. This means that, \[a=b=x\] . The number of soldiers which are in the square is $4770-9=4761$ . But, the number of soldiers in the square is also the number of rows times the number of columns. This means,
$\begin{align}
  & {{x}^{2}}=4761 \\
 & \Rightarrow x=\sqrt{4761}=69 \\
\end{align}$
Therefore, we can conclude that the number of soldiers in each row is $69$.

Note: The first mistake that students make in this problem is that they ignore the $9$ soldiers that are left out of the square and write ${{x}^{2}}=4770$ . Also, students may think that the soldiers are arranged along the perimeter of the square rather than filling it. Both the approaches are wrong and should be kept in mind while solving the problem.