Four years from now, the ratio of the ages of P and Q will be 3 : 5, respectively. The ratio of the present ages of R and Q is 5 : 6, respectively. If the sum of the ages of P and R six years from now is 62 years, what is the sum of the present ages of P and Q?

The correct option is 56 years.

Let us solve this step-by-step by defining variables for the present ages of P, Q, and R.

Step 1: Formulate equations using the given ratios.

Let the present ages of P, Q, and R be $P$, $Q$, and $R$ years, respectively.

The ratio of the present ages of R and Q is given as $5:6$.
Therefore, we can represent their present ages in terms of a variable $x$:
$R=5x$
$Q=6x$

Step 2: Use the information about ages 4 years from now.

Four years from now:
Age of P = $P+4$
Age of Q = $Q+4=6x+4$

The ratio of their ages 4 years from now is $3:5$:
$\frac{P+4}{6x+4}=\frac{3}{5}$

Cross-multiplying to express $P$ in terms of $x$:
$5(P+4)=3(6x+4)$
$5P+20=18x+12$
$5P=18x-8$
$P=\frac{18x-8}{5}$

Step 3: Use the condition for sum of ages 6 years from now.

Six years from now:
Age of P = $P+6$
Age of R = $R+6=5x+6$

The sum of their ages six years from now is given as 62 years:
$(P+6)+(5x+6)=62$
$P+5x+12=62$
$P+5x=50$

Step 4: Solve for $x$.

Substitute the expression for $P$ from Step 2 into this equation:
$\frac{18x-8}{5}+5x=50$

Multiply the entire equation by 5 to clear the fraction:
$18x-8+25x=250$
$43x-8=250$
$43x$=258
$x=\frac{258}{43}$
$x=6$

Step 5: Calculate the present ages of P and Q.

Present age of Q:
$Q=6x=6\times 6=36\text{years}$

Present age of P:
$P=\frac{18\left(6\right)-8}{5}=\frac{108-8}{5}=\frac{100}{5}=20\text{years}$

Step 6: Find the sum of the present ages of P and Q.

Sum of present ages of P and Q = $P+\mathrm{Q}=20+36=56\text{years}$

Thus, the sum of the present ages of P and Q is 56 years.