How Many Games Did the Team Play Overall?

Assuming every game ended in either a win or a loss, the team played 20 games overall. It had played 15 games before the five additional games.

The key is to track the number of losses, not just compare the two percentages. Initially, an 80% win rate means a 20% loss rate. After the added games, the team has two more losses and five more games in total.

Set up the starting total

Let $n$ be the number of games the team had played before the five new games. Since it won 80% of its games, it lost the remaining 20%, or $0.20n$ games.

The team then played five more games: it won three and lost two. So its loss count increased by 2, and its total number of games increased by 5. The new loss percentage is 25%, giving the equation

$$\frac{0.20n+2}{n+5}=0.25.$$

The numerator is the new number of losses; the denominator is the new total number of games.

Solve the equation

Multiply both sides by $n+5$:

$$\begin{aligned} 0.20n+2 &= 0.25(n+5) \\ 0.20n+2 &= 0.25n+1.25 \\ 0.75 &= 0.05n \\ n &= 15. \end{aligned}$$

So the team had played 15 games before the five additional games. Its overall total is

$$15+5=20.$$

Check with the game record

Out of the first 15 games, 80% were wins:

$$0.80\times 15=12 \text{ wins}.$$

That leaves 3 losses. After the additional games, the record is:

• Wins: $12+3=15$
• Losses: $3+2=5$
• Total games: $15+5=20$

The final loss percentage is $5/20=0.25$, or 25%, as stated. This also confirms that the initial number of games is consistent with a whole-number record.

Why the percentage changes

A percentage is a ratio, so both the number of losses and the total number of games matter. The additional games include two losses, which is a 40% loss rate for those five games. Since 40% is higher than the team’s original 20% loss rate, the overall loss percentage rises.

You cannot find the answer by simply subtracting 20% from 25% and treating that difference as a count of games. Percentages have different denominators here: the initial percentage uses $n$ games, while the later percentage uses $n+5$ games. The equation handles that denominator change directly.

Common mistakes

• Using wins instead of losses without adjusting the target. The final condition is about the loss percentage, so count losses in the numerator. You could instead use wins, but then you must use a 75% win percentage at the end.
• Adding five to the loss count. Only two of the new games were losses. The other three were wins; all five increase the total number of games.
• Using $n$ as the final total. The unknown $n$ is the number of games before the extra set. The final total is $n+5$.
• Assuming ties are included without checking. The result relies on the usual interpretation that every game is either a win or a loss. If ties are possible, an 80% win rate does not by itself establish that the initial loss rate was 20%; more information about ties would be needed.

If the stated percentages are rounded rather than exact, the information may not determine a unique game count. Under the usual exact-percentage interpretation, the answer is 20 games overall.