Chapter 7.1, Problem 55E

a.

To determine

To copy the board and mark the given changes.

Answer to Problem 55E

The completed chess board:

  

Explanation of Solution

Given:

The given $10\times 10$ boardis a chess board in which the dots represent the opponent’s chess pieces.

  

Calculation:

As per the problem

Consider the given chess board

There are $22$ dots that represent chess pieces, on the opponent’s side.

Mark $22$ ‘X’ on the other side in the same way as arranged on the opponent’s side and complete the chess board.

  

The $22$ ‘X’ represents chess pieces of the other player.

Conclusion:

The completed chess board:

  

b.

To determine

To calculate the percent of squares occupied in the given board.

Answer to Problem 55E

  $44\%$ of the squares are occupied on the board.

Explanation of Solution

Given:

The given board (as drawn in part (a.)) :

  

Calculation:

As per the given problem

Consider the board drawn in part (a)

  

Total number of squares = $100$

Number of squares occupied by chess pieces = $22+22=44\text{ [each side has 22 chess pieces]}$

Percent of squares occupied = $\frac{\text{number of squares occupied}}{\text{total number of squares}}\cdot 100$

  $\begin{array}{l}\Rightarrow \frac{44}{100}\cdot 100\\ \Rightarrow 44\%\end{array}$

Hence,

Percent of squares occupied on the board is $44\%$ .

Conclusion:

  $44\%$ of the squares are occupied on the board.

c.

To determine

To calculate the number of chess pieces out of the $75\%$ still on the board that belongs to the opponent.

Answer to Problem 55E

  $\text{19}$ chess pieces out of the $75\%$ still on the board belongs to the opponent.

Explanation of Solution

Given:

  $75\%$ of the chess pieces are still on the board.

Number of chess pieces out of $75\%$ that belongs to you = $14$ .

Calculation:

As per the given problem

Consider the table drawn in part (a.)

Number of chess pieces that were on the board primarily = $44$

  $75\%$ of the chess pieces are still on the board.

Number of chest pieces still on the board

  $\begin{array}{l}=75\%\cdot 44\\ =\frac{75}{100}\cdot 44\text{ [write percent as fraction]}\\ \text{= 33 [simplify]}\end{array}$

Number of chess pieces out of $75\%$ that belongs to you = $14$ .

Therefore, number of chest pieces that belong to the opponent

  $\text{=33-14=19}$

Conclusion:

  $\text{19}$ chess pieces out of the $75\%$ still on the board belongs to the opponent.