The average marks (out of 100) of boys and girls in an examination are 70 and 73 respectively. If the average marks of all the students in the examination is 71, then find the ratio of the number of boys to the number of girls.

Options

A. `1 : 3`
B. `2 : 1`
C. `1 : 2`
D. `3 : 1`

Answer

To solve the problem, we need to find the ratio of the number of boys to the number of girls based on their average marks and the overall average marks. Let's denote:

- Let the number of boys be \( x \)
- Let the number of girls be \( y \)

Given data:
- Average marks of boys = 70
- Average marks of girls = 73
- Average marks of all students = 71

### Step 1: Set up the equation for total marks
The total marks obtained by boys and girls can be expressed as:
- Total marks of boys = \( 70x \)
- Total marks of girls = \( 73y \)

The total marks of all students (boys + girls) is:
\[ \text{Total marks} = 70x + 73y \]

### Step 2: Set up the equation for the overall average
The overall average marks for all students can be expressed as:
\[ \text{Average} = \frac{\text{Total marks}}{\text{Total number of students}} = \frac{70x + 73y}{x + y} \]
According to the problem, this average is equal to 71:
\[ \frac{70x + 73y}{x + y} = 71 \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 70x + 73y = 71(x + y) \]

### Step 4: Expand the right side
Expanding the right side:
\[ 70x + 73y = 71x + 71y \]

### Step 5: Rearrange the equation
Rearranging the equation to isolate terms involving \( x \) and \( y \):
\[ 70x + 73y - 71x - 71y = 0 \]
This simplifies to:
\[ -1x + 2y = 0 \]

### Step 6: Solve for the ratio
Rearranging gives:
\[ x = 2y \]
Thus, the ratio of boys to girls is:
\[ \frac{x}{y} = \frac{2y}{y} = 2 \]
So, the ratio of boys to girls is:
\[ \text{Ratio of boys to girls} = 2:1 \]

### Final Answer
The ratio of the number of boys to the number of girls is \( 2:1 \).

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