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03 Aug 2026, 10:37
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A high school volleyball team completed its regular season last spring. If every match ended in either a win or a loss, with no ties, how many matches did the team play that season?
(1) If the team had lost 4 more matches than it actually lost, it would have won 25% of its matches.
(2) If the team had won 2 more matches than it actually won, it would have lost 50% of its matches.
(1) If the team had lost 4 more matches than it actually lost, it would have won 25% of its matches.
(2) If the team had won 2 more matches than it actually won, it would have lost 50% of its matches.
03 Aug 2026, 10:37
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Official Solution:
A high school volleyball team completed its regular season last spring. If every match ended in either a win or a loss, with no ties, how many matches did the team play that season?
Let \(W\) be the number of wins and \(L\) the number of losses. Then total matches \(= W + L\).
(1) If the team had lost 4 more matches than it actually lost, it would have won \(25\%\) of its matches.
If the team had lost 4 more matches, it would have won 4 fewer matches. So the number of wins would have been \(W - 4\). The statement says that then the team would have won \(25\%\) of its matches, so:
\(W - 4 = 0.25(W + L)\)
\(3W - L = 16\)
This alone is not enough to find \(W + L\).
Not sufficient.
(2) If the team had won 2 more matches than it actually won, it would have lost \(50\%\) of its matches.
If the team had won 2 more matches, it would have lost 2 fewer matches. So the number of losses would have been \(L - 2\). The statement says that then the team would have lost \(50\%\) of its matches, so:
\(L - 2 = 0.5(W + L)\)
\(L - W = 4\)
This alone is not enough to find \(W + L\).
Not sufficient.
(1)+(2) When combining, we have two distinct linear equations with two unknowns, \(3W - L = 16\) and \(L - W = 4\). We can solve for \(W\) and \(L\) and get the value of \(W + L\). Sufficient.
Answer: C
A high school volleyball team completed its regular season last spring. If every match ended in either a win or a loss, with no ties, how many matches did the team play that season?
Let \(W\) be the number of wins and \(L\) the number of losses. Then total matches \(= W + L\).
(1) If the team had lost 4 more matches than it actually lost, it would have won \(25\%\) of its matches.
If the team had lost 4 more matches, it would have won 4 fewer matches. So the number of wins would have been \(W - 4\). The statement says that then the team would have won \(25\%\) of its matches, so:
\(W - 4 = 0.25(W + L)\)
\(3W - L = 16\)
This alone is not enough to find \(W + L\).
Not sufficient.
(2) If the team had won 2 more matches than it actually won, it would have lost \(50\%\) of its matches.
If the team had won 2 more matches, it would have lost 2 fewer matches. So the number of losses would have been \(L - 2\). The statement says that then the team would have lost \(50\%\) of its matches, so:
\(L - 2 = 0.5(W + L)\)
\(L - W = 4\)
This alone is not enough to find \(W + L\).
Not sufficient.
(1)+(2) When combining, we have two distinct linear equations with two unknowns, \(3W - L = 16\) and \(L - W = 4\). We can solve for \(W\) and \(L\) and get the value of \(W + L\). Sufficient.
Answer: C
