The average marks of 50 students of a class is 76. If the average marks of all boys is 70 and that of all girls is 80 in that class, then find the number of boys in the class.

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the total marks of all students
The average marks of 50 students is 76. Therefore, the total marks of all students can be calculated using the formula:

\[ \text{Total Marks} = \text{Average Marks} \times \text{Number of Students} \]

Substituting the values:

\[ \text{Total Marks} = 76 \times 50 = 3800 \]

### Step 2: Let the number of boys be \( b \) and the number of girls be \( g \)
Since the total number of students is 50, we can write:

\[ b + g = 50 \]

### Step 3: Calculate the total marks of boys and girls
The average marks of boys is 70, so the total marks of boys can be expressed as:

\[ \text{Total Marks of Boys} = 70b \]

The average marks of girls is 80, so the total marks of girls can be expressed as:

\[ \text{Total Marks of Girls} = 80g \]

### Step 4: Write the equation for total marks
The total marks of all students can also be expressed as the sum of the total marks of boys and girls:

\[ 70b + 80g = 3800 \]

### Step 5: Substitute \( g \) from the first equation into the second equation
From the equation \( b + g = 50 \), we can express \( g \) as:

\[ g = 50 - b \]

Now, substitute this into the total marks equation:

\[ 70b + 80(50 - b) = 3800 \]

### Step 6: Simplify the equation
Expanding the equation gives:

\[ 70b + 4000 - 80b = 3800 \]

Combine like terms:

\[ -10b + 4000 = 3800 \]

### Step 7: Solve for \( b \)
Rearranging the equation gives:

\[ -10b = 3800 - 4000 \]

\[ -10b = -200 \]

Dividing both sides by -10:

\[ b = 20 \]

### Step 8: Find the number of girls
Using the equation \( g = 50 - b \):

\[ g = 50 - 20 = 30 \]

### Final Answer
The number of boys in the class is **20**.

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