In a class, the number of boys and girls are distinct. The average age of all the student in the class equals the average of the average age of the boys and the average age of the girls. The sum of the average age of the boys and the average age of the girls is 10 years. Find the average age (in years) of the boys of the class

Answer

To solve the problem step by step, we can follow this reasoning:

### Step 1: Understand the Given Information
We know that:
- The average age of all students in the class is equal to the average of the average age of boys and girls.
- The sum of the average age of boys (B) and the average age of girls (G) is 10 years.

### Step 2: Set Up the Equation
From the information given, we can write:
\[ \frac{B + G}{2} = \text{Average age of the class} \]
And we also know:
\[ B + G = 10 \]

### Step 3: Express Average Age of the Class
From the equation \( B + G = 10 \), we can express G in terms of B:
\[ G = 10 - B \]

### Step 4: Substitute G in the Average Age Equation
Now, substitute G in the average age equation:
\[ \frac{B + (10 - B)}{2} = \text{Average age of the class} \]
This simplifies to:
\[ \frac{10}{2} = 5 \]
So, the average age of the class is 5 years.

### Step 5: Analyze the Implications of the Average
Since the average age of the class is 5 years, and the average of the boys and girls must be distinct (as stated in the problem), we know that:
- The average age of boys (B) and the average age of girls (G) must be such that one is greater than 5 and the other is less than 5.

### Step 6: Determine Possible Values for B and G
Given that \( B + G = 10 \) and one must be less than 5 while the other is greater than 5, we can conclude:
- If \( G < 5 \), then \( B > 5 \).
- If \( G = 5 \), then \( B = 5 \), which contradicts the distinctness condition.

### Step 7: Conclude the Average Age of Boys
Since the average age of boys must be greater than 5, and the average age of girls must be less than 5, we can conclude:
- The only possible average age for boys that satisfies all conditions is \( B = 5 \).

Thus, the average age of the boys in the class is **5 years**.