Our badminton team has finished $75\%$ of its season. So far, we have won $32\%$ of the games we played. What percent of the remainder of our games must we win in order to finish the season with the same number of wins as losses?
If the team has finished $75\%$ of its season, then $100\% - 75\% = 25\%$ of the season remains. If $32\%$ of the games played has been won, then the team has lost $100\% - 32\% = 68\%$ of the games they played so far. Since the team has played $75\%$ of the games when it has lost $68\%$ of the games played so far, if $(100\%-75\%)$ percent of the games remain, $(100\%-68\%)$ percent of the remaining games must be won in order to finish with the same number of wins as losses. Therefore, $25\% \cdot \frac{100\%-68\%}{100\%-75\%}=\boxed{40\%}$ of the remainder of the games must be won.
Let's start by finding the percentage of games played by subtracting $75\%$ from $100\%$: $100\% - 75\% = 25\%$.
Next, we need to find the percentage of games we have lost. Since we have won $32\%$ of the games, the percentage of games lost is $100\% - 32\% = 68\%$.
Since we want to finish the season with the same number of wins as losses, we need to win the same percentage of the remaining games as we have lost so far.
Since $68\%$ of the games played have been lost, we need to win $68\%$ of the remaining $25\%$ of the games.
Therefore, we need to win $\frac{68\%}{100\%}\times\frac{25\%}{100\%} = \frac{17\%}{100\%} = 17\%$ of the remainder of our games to finish the season with the same number of wins as losses.