Question

The area of each square in a chessboard is \[4\] sq.cm. Find the area of the board.
a) At the beginning of the game when all the chessmen are put on the board, write the area of the squares left unoccupied.
b) Find the area of the squares occupied by the chessmen.

Answer 1

Hint: Total area of the chessboard can be found using area of each square and the total number of squares. Area of the squares left unoccupied can be obtained using the number of squares left unoccupied. The remaining area that is the area of the squares occupied can be found by the difference of the above values.

Formula used: If the area of a square (or whatever object) is $A$ and if there are $n$ squares of that kind, then the total area is the product $nA$.
Number of squares in a standard chessboard is ${8^2} = 64$

Complete step-by-step answer:
Given that the area of each square in a chessboard, $a$ is $4c{m^2}$.
Total number of squares in a chessboard, $n$ is ${8^2} = 64$.
So the area of the chessboard, $A$ equals the product of the number of squares and area of each square.
$A = n \times a = 64 \times 4 = 256c{m^2}$
a) At the beginning of the game some of the squares are occupied and some are unoccupied.
Number of squares occupied by a player is $16$.
Number of players in a chess game is $2$.
Therefore, the number of squares occupied by two players ${n_1}$ is $2 \times 16 = 32$.
Number of squares unoccupied, ${n_2}$$ = $total number of squares$ - $number of squares occupied.
$ \Rightarrow {n_2} = 64 - {n_1} = 64 - 32 = 32$
Area of the squares unoccupied ${A_2} = $ number of squares unoccupied $ \times $ area of each square
${A_2} = {n_2} \times a = 32 \times 4c{m^2} = 128c{m^2}$
b) Area of the squares occupied by the chessmen, ${A_1} = $ Total area of the chessboard $ - $ Area of the squares unoccupied
$ \Rightarrow {A_1} = A - {A_2} = 256 - 128 = 128c{m^2}$
Total area of the chessboard $ = 256c{m^2}$
Area of the squares unoccupied $ = 128c{m^2}$
Area of the squares occupied $ = 128c{m^2}$

Note: In a chessboard total number of small squares is $64$. But the overall number of squares is not equal to $64$. Here we consider squares of all sizes. That is $64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204$. So we must be careful about this.