The sum of the present ages of P & Q and R & Q is 64 years and 56 years, respectively. The ratio of the age of P after 12 years to the age of Q six years ago is 4:X. If the age of R after 12 years is 32 years, then find X.

To find the value of $X$, we can break down the problem step-by-step using algebraic equations based on the given information.

Step 1: Write down the equations for the sum of their present ages.
Let the present ages of P, Q, and R be $P$, $Q$, and $R$ years respectively.
According to the problem, the sum of the present ages of P and Q is 64 years:
$P+Q=64$ — (Equation 1)
Similarly, the sum of the present ages of R and Q is 56 years:
$R+Q=56$ — (Equation 2)

Step 2: Find the present age of R.
We are given that the age of R after 12 years is 32 years. This can be expressed as:
$R+12=32$
Subtracting 12 from both sides, we get the present age of R:
$R=32-12=20$ years.

Step 3: Calculate the present age of Q.
Substitute the value of R ($R=20$) into Equation 2:
$20+Q=56$
Subtracting 20 from both sides gives the present age of Q:
$Q=56-20=36$ years.

Step 4: Calculate the present age of P.
Substitute the value of Q ($Q=36$) into Equation 1:
$P+36=64$
Subtracting 36 from both sides gives the present age of P:
$P=64-36=28$ years.

Step 5: Determine the ages required for the ratio.
The age of P after 12 years is:
$P+12=28+12=40$ years.
The age of Q six years ago is:
$Q-6=36-6=30$ years.

Step 6: Solve for X using the given ratio.
The ratio of the age of P after 12 years to the age of Q six years ago is given as 4:X:
$\frac{40}{30}=\frac{4}{X}$
Simplifying the left-hand side:
$\frac{4}{3}=\frac{4}{X}$
By comparing the numerators, we find:
$X=3$

Thus, the correct option is 3.