The mean marks of boys & girls in periodical test are 36 and 39 respectively. If the mean marks of all the students of class IX in that test are 37. Find the ratio of the number of boys to the number of girls.

Answer

To solve the problem, we will follow these steps:

### Step 1: Define Variables
Let:
- \( X \) = number of boys
- \( Y \) = number of girls
- \( A \) = sum of marks of boys
- \( B \) = sum of marks of girls

### Step 2: Write the Mean Formulas
From the information given:
- Mean marks of boys = \( \frac{A}{X} = 36 \)
- Mean marks of girls = \( \frac{B}{Y} = 39 \)
- Mean marks of all students = \( \frac{A + B}{X + Y} = 37 \)

### Step 3: Express A and B in terms of X and Y
From the mean formulas, we can express \( A \) and \( B \):
1. \( A = 36X \) (from the boys' mean)
2. \( B = 39Y \) (from the girls' mean)

### Step 4: Substitute A and B into the Total Mean Formula
Now substitute \( A \) and \( B \) into the total mean formula:
\[ \frac{A + B}{X + Y} = 37 \]
Substituting the values of \( A \) and \( B \):
\[ \frac{36X + 39Y}{X + Y} = 37 \]

### Step 5: Cross Multiply to Eliminate the Denominator
Cross multiplying gives:
\[ 36X + 39Y = 37(X + Y) \]

### Step 6: Expand and Rearrange the Equation
Expanding the right side:
\[ 36X + 39Y = 37X + 37Y \]
Rearranging the equation:
\[ 36X + 39Y - 37X - 37Y = 0 \]
This simplifies to:
\[ -1X + 2Y = 0 \]
or:
\[ X = 2Y \]

### Step 7: Find the Ratio of Boys to Girls
The ratio of the number of boys to the number of girls is:
\[ \frac{X}{Y} = \frac{2Y}{Y} = 2 \]
Thus, the ratio of boys to girls is:
\[ \text{Ratio} = 2:1 \]

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

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