The mean age of three individuals, P, Q, and R, is 54 years. The ratio of the present ages of P to R is 4 : 9, while Q's present age is 25% greater than that of P. What is the difference between the age of P after 7 years and the age of Q 5 years ago?

The correct option is 3 years.

Given Data:
1. Mean age of P, Q, and R = 54 years
2. Ratio of present ages of P to R = 4 : 9
3. Present age of Q is 25% greater than that of P

Step 1: Find the sum of the present ages of P, Q, and R

$\text{Sum of present ages}=\text{Mean age}\times 3$

$\text{Sum of present ages}=54\times 3=162\text{years}$

Step 2: Express the present ages of P, Q, and R in terms of a common ratio variable $x$

From the given ratio of P to R (4 : 9):
Let the present age of P = $4x$
Let the present age of R = $9x$

Since Q's present age is 25% greater than P's present age:

$\text{Age of Q}=4x+\left(25\%\text{of}4x\right)$

$\text{Age of Q}=4x+\left(0.25\times 4x\right)=4x+x=5x$

Step 3: Solve for $x$

$\text{Sum of ages}=\text{Age of P}+\text{Age of Q}+\text{Age of R}$

$4x+5x+9x=162$

$18x=162$

$x=\frac{162}{18}=9$

Step 4: Calculate the present ages of P and Q

$\text{Present age of P}=4\times 9=36\text{years}$

$\text{Present age of Q}=5\times 9=45\text{years}$

Step 5: Find the required difference

$\text{Age of P after 7 years}=36+7=43\text{years}$

$\text{Age of Q 5 years ago}=45-5=40\text{years}$

$\text{Difference}=43-40=3\text{years}$

Therefore, the difference between the age of P after 7 years and the age of Q 5 years ago is 3 years.