A professional gambler has won `40%` of his 25 poker games for the week so far. If, all of a sudden, his luck changes and he begins winning `80%` of the time, how many more games must he play to end up winning `60%` of all his games for the week?

Answer

To solve the problem step by step, let's break it down:

### Step 1: Determine the number of games won so far
The gambler has won 40% of his 25 poker games. We can calculate the number of games won as follows:

\[ \text{Games won} = 0.4 \times 25 = 10 \]

**Hint:** To find the number of games won, multiply the total games played by the winning percentage.

### Step 2: Set up the equation for future games
Let \( x \) be the number of additional games the gambler needs to play. If he wins 80% of these games, the number of games won in these additional games will be:

\[ \text{Games won in additional games} = 0.8 \times x \]

### Step 3: Calculate total games played and total games won
After playing \( x \) more games, the total number of games played will be:

\[ \text{Total games played} = 25 + x \]

The total number of games won after playing these additional games will be:

\[ \text{Total games won} = 10 + 0.8x \]

### Step 4: Set up the equation for the desired winning percentage
We want the gambler to have won 60% of all his games. Therefore, we can set up the equation:

\[ \frac{10 + 0.8x}{25 + x} = 0.6 \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:

\[ 10 + 0.8x = 0.6(25 + x) \]

### Step 6: Expand the equation
Expanding the right side:

\[ 10 + 0.8x = 15 + 0.6x \]

### Step 7: Rearrange the equation
Rearranging the equation to isolate \( x \):

\[ 10 + 0.8x - 0.6x = 15 \]

This simplifies to:

\[ 0.2x = 15 - 10 \]

\[ 0.2x = 5 \]

### Step 8: Solve for \( x \)
Dividing both sides by 0.2 gives:

\[ x = \frac{5}{0.2} = 25 \]

### Conclusion
The gambler must play **25 more games** to achieve a winning percentage of 60%.

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