In a school, `5/12` of the number of students are girls and the rest are boys, `4/7` of the number of boys are below 14 years of age, and `2/5` of the number of girls are 14 years or above 14 years of age. If the number of students below 14 years of age is 1120, then the total number in the school is:
एक स्कूल में, छात्रों की संख्या में से `5/12` लड़कियाँ हैं और बाकी लड़के हैं, जिनमें से `4/7` लड़कों की संख्या 14 वर्ष से कम है, और `2/5` की संख्या में लड़कियाँ 14 वर्ष या 14 वर्ष से अधिक हैं। यदि 14 वर्ष से कम आयु के छात्रों की संख्या 1120 है, तो स्कूल में कुल संख्या है:

Options

A. `1820`
B. `1900`
C. `1920`
D. `1290`

Answer

To solve the problem step by step, let's denote the total number of students in the school as \( x \).

### Step 1: Determine the number of girls and boys
According to the problem, \( \frac{5}{12} \) of the students are girls. Therefore, the number of girls is:
\[ \text{Number of girls} = \frac{5}{12} x \]
The remaining students are boys, which can be calculated as:
\[ \text{Number of boys} = x - \frac{5}{12} x = \frac{7}{12} x \]

### Step 2: Calculate the number of boys below 14 years of age
We know that \( \frac{4}{7} \) of the boys are below 14 years of age. Thus, the number of boys below 14 years is:
\[ \text{Boys below 14} = \frac{4}{7} \left(\frac{7}{12} x\right) = \frac{4}{12} x = \frac{1}{3} x \]

### Step 3: Calculate the number of girls above 14 years of age
The problem states that \( \frac{2}{5} \) of the girls are 14 years or above. Therefore, the number of girls above 14 years is:
\[ \text{Girls above 14} = \frac{2}{5} \left(\frac{5}{12} x\right) = \frac{2}{12} x = \frac{1}{6} x \]

### Step 4: Calculate the number of girls below 14 years of age
To find the number of girls below 14 years of age, we subtract the number of girls above 14 from the total number of girls:
\[ \text{Girls below 14} = \frac{5}{12} x - \frac{1}{6} x \]
To perform the subtraction, we need a common denominator:
\[ \frac{1}{6} x = \frac{2}{12} x \]
Thus,
\[ \text{Girls below 14} = \frac{5}{12} x - \frac{2}{12} x = \frac{3}{12} x = \frac{1}{4} x \]

### Step 5: Total number of students below 14 years of age
The total number of students below 14 years of age is the sum of boys and girls below 14:
\[ \text{Total below 14} = \text{Boys below 14} + \text{Girls below 14} \]
Substituting the values we calculated:
\[ \text{Total below 14} = \frac{1}{3} x + \frac{1}{4} x \]
To add these fractions, we need a common denominator, which is 12:
\[ \frac{1}{3} x = \frac{4}{12} x \quad \text{and} \quad \frac{1}{4} x = \frac{3}{12} x \]
Thus,
\[ \text{Total below 14} = \frac{4}{12} x + \frac{3}{12} x = \frac{7}{12} x \]

### Step 6: Set up the equation
We know from the problem that the total number of students below 14 years of age is 1120:
\[ \frac{7}{12} x = 1120 \]

### Step 7: Solve for \( x \)
To find \( x \), we can multiply both sides by \( \frac{12}{7} \):
\[ x = 1120 \times \frac{12}{7} \]
Calculating this gives:
\[ x = 1120 \times \frac{12}{7} = 1120 \times 1.7142857142857143 \approx 2400 \]

### Conclusion
The total number of students in the school is:
\[ \boxed{2400} \]