In a class `20%` of students are below 14 years of age . Out of the remaining students `10%` are of the age 14 - 15years and ratio of students who are between 15 - 16 years of age to student above 16 years of age is 3 : 2 . If the number of students who are above 16 years is 72 , what is the total number of students in the class ?

Options

A. 200
B. 250
C. 300
D. 400

Answer

To solve the problem step by step, we will follow the information provided in the question carefully.

### Step 1: Define the total number of students
Let the total number of students in the class be \( n \).

### Step 2: Calculate the number of students below 14 years of age
According to the question, 20% of the students are below 14 years of age. Therefore, the number of students below 14 years is:
\[ \text{Number of students below 14} = 0.2n \]

### Step 3: Calculate the remaining students
The remaining students after excluding those below 14 years of age will be:
\[ \text{Remaining students} = n - 0.2n = 0.8n \]

### Step 4: Calculate the number of students aged 14 to 15 years
Out of the remaining students, 10% are aged between 14 to 15 years. Thus, the number of students aged 14 to 15 years is:
\[ \text{Number of students aged 14 to 15} = 0.1 \times 0.8n = 0.08n \]

### Step 5: Calculate the number of students aged 15 to 16 years and above 16 years
Let \( x \) be the number of students aged between 15 to 16 years. The ratio of students aged 15 to 16 years to those above 16 years is given as 3:2. We also know that the number of students above 16 years is 72. Therefore, we can set up the following relationship:
\[ \frac{x}{72} = \frac{3}{2} \]

### Step 6: Solve for \( x \)
Cross-multiplying gives:
\[ 2x = 3 \times 72 \]
\[ 2x = 216 \]
\[ x = \frac{216}{2} = 108 \]

### Step 7: Calculate the total number of students
Now we can sum up all the students:
- Students below 14 years: \( 0.2n \)
- Students aged 14 to 15 years: \( 0.08n \)
- Students aged 15 to 16 years: \( 108 \)
- Students above 16 years: \( 72 \)

Setting up the equation:
\[ n = 0.2n + 0.08n + 108 + 72 \]
\[ n = 0.28n + 180 \]

### Step 8: Solve for \( n \)
Rearranging gives:
\[ n - 0.28n = 180 \]
\[ 0.72n = 180 \]
\[ n = \frac{180}{0.72} = 250 \]

### Conclusion
The total number of students in the class is \( \boxed{250} \).