Question

A general wishing to draw up his 64019 men in the form of a square found that he had 10 men extra. Find the number of men in the front row.

Answer 1

Hint: The total number of people is 64019 and out of which 10 persons remain to be arranged. Now, calculate the actual number of people. Assume that the number of people arranged in the front row is x. As the people are arranged in the form of a square, the number of rows will be the same as the number of people in the front row. Since x people are arranged in x number of rows, the total number of people arranged will be the product of the number of people in the front row and the number of rows. The number of people arranged in the form of the square is equal to \[{{x}^{2}}\] . But, we already have calculated the actual number of people. Now, compare the actual number of people and \[{{x}^{2}}\] . Then, solve it further and get the value of x.

Complete step by step solution:
According to the question, it is given that a general wishing to draw up his 64019 men in the form of a square found that he had 10 men extra. It means that after arranging the men in the form of a square, 10 men are left to be arranged.
The total number of men to be arranged in the form of a square = 64019 ……………………….(1)
The number of men left to be arranged = 10 …………………………(2)
The number of people arranged in the form of the square = \[64019-10=64009\] ……………………………(3)
Let us assume that the number of people arranged in the front row is x.
The number of people arranged in the first row = x …………………………………(4)
As the people are arranged in the form of a square so, the number of rows will be the same as the number of people in the front row.
From equation (4), we have the number of people arranged in the front row.
Since x people are arranged in x number of rows so, the total number of people arranged will be the product of the number of people in the front row and the number of rows.
The number of people arranged in the form of the square = \[x\times x={{x}^{2}}\] ………………………………………..(5)
From equation (3), we also have the number of people arranged in the form of the square.
On, comparing equation (3) and equation (5), we get
\[\begin{align}
  & \Rightarrow {{x}^{2}}=64009 \\
 & \Rightarrow x=\sqrt{64009} \\
\end{align}\]
\[\Rightarrow x=253\]
Hence, the number of people standing in the first row is 253.

Note: In this question, one might miss the point that 10 people are left who are not arranged. One might take the 64010 as the number of people arranged in the form of the square. This is wrong because out of 64019 people, 10 people are left to be arranged. So, the number of people arranged in the form of a square is 64009.