In the International Oxford School, all students play at least one of the two games rugby and baseball . 40% of all students play both rugby and baseball. If 20% of the students who play baseball do not play rugby , then what is the percentage of all students who plya baseball ?

Answer

To solve the problem step by step, we will follow a structured approach:

### Step 1: Define the Total Number of Students
Assume the total number of students in the International Oxford School is 100. This assumption simplifies the calculations.

**Hint:** Start with a manageable number to make percentage calculations easier.

### Step 2: Identify the Students Playing Both Games
According to the problem, 40% of all students play both rugby and baseball. Therefore, the number of students who play both games is:
\[ 40\% \text{ of } 100 = 40 \text{ students} \]

**Hint:** Convert percentages to actual numbers using the total assumed.

### Step 3: Define Variables for Students Playing Only One Game
Let \( x \) be the number of students who play only baseball. Consequently, the number of students who play only rugby can be calculated as:
\[ \text{Students playing only rugby} = 100 - 40 - x = 60 - x \]

**Hint:** Use variables to represent unknown quantities for clarity.

### Step 4: Calculate Total Students Playing Baseball
The total number of students who play baseball (including those who play both games) is:
\[ \text{Total baseball players} = x + 40 \]

**Hint:** Break down the total into parts to understand the composition.

### Step 5: Use Given Information About Baseball Players
The problem states that 20% of the students who play baseball do not play rugby. This means:
\[ x = 20\% \text{ of } (x + 40) \]
Expressing this mathematically:
\[ x = \frac{20}{100} \times (x + 40) \]
This simplifies to:
\[ x = \frac{1}{5} (x + 40) \]

**Hint:** Set up equations based on the relationships described in the problem.

### Step 6: Solve the Equation
Multiply both sides by 5 to eliminate the fraction:
\[ 5x = x + 40 \]
Rearranging gives:
\[ 5x - x = 40 \]
Thus:
\[ 4x = 40 \]
Dividing both sides by 4 results in:
\[ x = 10 \]

**Hint:** Isolate the variable to find its value.

### Step 7: Calculate Total Students Playing Baseball
Now that we have \( x \):
\[ \text{Total baseball players} = x + 40 = 10 + 40 = 50 \]

**Hint:** Combine the parts to find the total.

### Step 8: Find the Percentage of Students Playing Baseball
To find the percentage of all students who play baseball:
\[ \text{Percentage of students playing baseball} = \left(\frac{50}{100}\right) \times 100\% = 50\% \]

**Hint:** Convert the total number of players back into a percentage of the total.

### Final Answer
The percentage of all students who play baseball is **50%**.

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