The ages of the students of a class form an A.P. whose common difference is 4 months. If the youngest student is 8 years old and the sum of the ages of all the students of the class is 168 years, find the number of students in the class.

Answer

To solve the problem, we need to find the number of students in a class whose ages form an arithmetic progression (A.P.). Here are the steps to find the solution:

### Step 1: Identify the parameters of the A.P.
- The youngest student is 8 years old, which we can denote as \( a = 8 \) years.
- The common difference \( d = 4 \) months. We need to convert this into years for consistency:
\[ d = \frac{4}{12} = \frac{1}{3} \text{ years} \]

### Step 2: Use the formula for the sum of the first \( n \) terms of an A.P.
The formula for the sum \( S_n \) of the first \( n \) terms of an A.P. is given by:
\[ S_n = \frac{n}{2} \left( 2a + (n-1)d \right) \]
We know that the sum of the ages of all the students is 168 years, so we can set up the equation:
\[ 168 = \frac{n}{2} \left( 2(8) + (n-1)\left(\frac{1}{3}\right) \right) \]

### Step 3: Simplify the equation
Substituting \( a = 8 \) and \( d = \frac{1}{3} \):
\[ 168 = \frac{n}{2} \left( 16 + \frac{n-1}{3} \right) \]
Multiply both sides by 2 to eliminate the fraction:
\[ 336 = n \left( 16 + \frac{n-1}{3} \right) \]
Now, multiply through by 3 to eliminate the fraction:
\[ 1008 = 3n \left( 16 + \frac{n-1}{3} \right) \]
This simplifies to:
\[ 1008 = 48n + n(n-1) \]
Rearranging gives:
\[ n^2 - n + 48n - 1008 = 0 \]
\[ n^2 + 47n - 1008 = 0 \]

### Step 4: Solve the quadratic equation
We can use the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = 47, c = -1008 \):
\[ b^2 - 4ac = 47^2 - 4 \times 1 \times (-1008) = 2209 + 4032 = 6241 \]
Now, calculate \( n \):
\[ n = \frac{-47 \pm \sqrt{6241}}{2 \times 1} = \frac{-47 \pm 79}{2} \]
Calculating the two possible values:
1. \( n = \frac{32}{2} = 16 \)
2. \( n = \frac{-126}{2} = -63 \) (not valid since number of students cannot be negative)

### Step 5: Conclusion
Thus, the number of students in the class is:
\[ \boxed{16} \]