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

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**Solution:**

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

2. According to the problem, four years later, the ratio of ages of P and Q is $3 : 5$:
   $$\frac{P + 4}{Q + 4} = \frac{3}{5}$$
   Cross-multiplying:
   $$5(P + 4) = 3(Q + 4)$$
   $$5P + 20 = 3Q + 12$$
   Rearranged:
   $$5P - 3Q = -8 \quad \quad (1)$$

3. The present ratio of ages of R and Q is $5 : 6$:
   $$\frac{R}{Q} = \frac{5}{6} \implies R = \frac{5}{6}Q \quad \quad (2)$$

4. After six years, the sum of ages of P and R is 62:
   $$(P + 6) + (R + 6) = 62$$
   $$P + R + 12 = 62$$
   $$P + R = 50 \quad \quad (3)$$

5. Substitute $R$ from equation (2) into equation (3):
   $$P + \frac{5}{6}Q = 50$$
   Multiply both sides by 6:
   $$6P + 5Q = 300 \quad \quad (4)$$

6. From equation (1), multiply both sides by 1 to keep it as is:
   $$5P - 3Q = -8$$

7. Now solve equations (4) and (1) simultaneously:
   - Multiply equation (1) by 5:
     $$25P - 15Q = -40$$
   - Multiply equation (4) by 3:
     $$18P + 15Q = 900$$

8. Add the two equations:
   $$25P - 15Q + 18P + 15Q = -40 + 900$$
   $$43P = 860$$
   $$P = \frac{860}{43} = 20$$

9. Substitute $P = 20$ into equation (1):
   $$5(20) - 3Q = -8$$
   $$100 - 3Q = -8$$
   $$-3Q = -108$$
   $$Q = 36$$

10. The sum of present ages of P and Q is:
    $$P + Q = 20 + 36 = 56$$

**Answer:** 56 years

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