The present age of P is twelve years less than that of Q. After four years, the ratio of the ages of P to Q will be $2:5$. If the present age of R is six years more than the age of Q two years ago, then find the difference between the present ages of R and P.

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**Solution:**

1. Let the present age of Q be $x$ years.

2. Since P's present age is twelve years less than Q's, we have:
   $$P = x - 12$$

3. After 4 years, the ages will be:
   - P's age: $(x - 12) + 4 = x - 8$
   - Q's age: $x + 4$

4. According to the problem, the ratio of their ages after 4 years is $2:5$:
   $$\frac{x - 8}{x + 4} = \frac{2}{5}$$

5. Cross-multiplied:
   $$5(x - 8) = 2(x + 4)$$
   $$5x - 40 = 2x + 8$$

6. Rearranging terms:
   $$5x - 2x = 8 + 40$$
   $$3x = 48$$
   $$x = 16$$

7. So, present age of Q is $16$ years.

8. Present age of P:
   $$P = 16 - 12 = 4 	ext{ years}$$

9. Present age of R is six years more than Q's age two years ago:
   $$R = (16 - 2) + 6 = 14 + 6 = 20 	ext{ years}$$

10. The difference between R's and P's present ages:
    $$20 - 4 = 16 	ext{ years}$$

**Answer:** $16$ years

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