1. **State the problem:** A father's age is 4 times that of his elder son and 5 times that of his younger son. When the elder son has lived three times his present age, the father's age will exceed twice that of the younger son by 4 years. We need to find their present ages. 2. **Define variables:** Let the elder son's present age be $x$ years and the younger son's present age be $y$ years. 3. **Express the father's age:** Since the father is 4 times the elder son's age and 5 times the younger son's age, we have two expressions for the father's age: $$F = 4x$$ $$F = 5y$$ 4. **Equate the father's age expressions:** Since both represent the same father's age, $$4x = 5y$$ 5. **Condition when elder son has lived three times his present age:** The elder son's age then is $3x$. At that time, the father's age will be $F + (3x - x) = F + 2x$ because the father also ages by the same amount of time passed. 6. **Condition for father's age exceeding twice the younger son's age by 4 years:** At that time, the younger son's age will be $y + 2x$ (since the same time passed). The condition is: $$F + 2x = 2(y + 2x) + 4$$ 7. **Substitute $F = 4x$ into the equation:** $$4x + 2x = 2(y + 2x) + 4$$ $$6x = 2y + 4x + 4$$ 8. **Simplify:** $$6x - 4x = 2y + 4$$ $$2x = 2y + 4$$ 9. **Divide both sides by 2:** $$x = y + 2$$ 10. **Recall from step 4:** $$4x = 5y$$ Substitute $x = y + 2$: $$4(y + 2) = 5y$$ $$4y + 8 = 5y$$ 11. **Solve for $y$:** $$8 = 5y - 4y$$ $$8 = y$$ 12. **Find $x$:** $$x = y + 2 = 8 + 2 = 10$$ 13. **Find father's age:** $$F = 4x = 4 \times 10 = 40$$ **Final answer:** - Elder son's present age: $10$ years - Younger son's present age: $8$ years - Father's present age: $40$ years