The ratio of the ages of two boys is 3:7. After 2 years, the ratio of their ages will become 5:9. The ratio of their ages after 10 years will be:

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Let the present ages of the two boys be $$ 3x $$ and $$ 7x $$.

1. **Given condition after 2 years:**  
   After 2 years, the ratio of their ages becomes $$ \frac{5}{9} $$:  
   $$
   \frac{3x + 2}{7x + 2} = \frac{5}{9}.
   $$

2. **Solve for $$ x $$:**  
   Cross-multiply to eliminate the fraction:  
   $$
   9(3x + 2) = 5(7x + 2).
   $$  
   Expand and simplify:  
   $$
   27x + 18 = 35x + 10.
   $$  
   Rearrange terms:  
   $$
   8x = 8 \implies x = 1.
   $$

3. **Find their present ages:**  
   - First boy's age = $$ 3x = 3(1) = 3 $$.  
   - Second boy's age = $$ 7x = 7(1) = 7 $$.

4. **Find their ages after 10 years:**  
   - First boy's age after 10 years = $$ 3 + 10 = 13 $$.  
   - Second boy's age after 10 years = $$ 7 + 10 = 17 $$.

5. **Ratio of their ages after 10 years:**  
   The required ratio is:  
   $$
   \frac{13}{17} = 13:17.
   $$

Thus, the ratio of their ages after 10 years is **13:17**, which corresponds to option **(d)**.

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