When their son was born, the ratio of the ages of a father and a mother was 11 : 10. When the son becomes twice his present age in the future, the ratio of the ages of the father and the mother will be 19 : 18. What is the ratio of the present ages of the father and the mother?

Introduction / Context:
This problem involves ratios of ages at different points in time and an indirect role of the son age. It is a classic example of how age ratio questions use past and future information together. The goal is to convert the given ratio statements into equations and eliminate the son age to get a direct ratio between the present ages of the father and the mother.

Given Data / Assumptions:

• When the son was born, the ratio of the ages of the father and mother was 11 : 10.
• In the future, when the son becomes twice his present age, the ratio of the ages of the father and mother will be 19 : 18.
• We must find the ratio of the present ages of the father and the mother.
• Ages are measured in years and assumed to be positive.

Concept / Approach:
The key idea is to introduce variables for present ages and relate them to ages at different times. At the time of birth of the son, both parents were younger by an amount equal to the son current age. At the future time when the son doubles his age, both parents are older by the same number of years. Both conditions are expressed as ratios that can be converted into algebraic equations. Solving these equations simultaneously allows us to determine the ratio of present ages of the parents without needing their exact ages.

Step-by-Step Solution:
Step 1: Let the present age of the father be F years and the present age of the mother be M years. Let the present age of the son be S years. Step 2: At the time of the son birth, the son age was 0 and both parents were younger by S years, so their ages were F - S and M - S. Step 3: At that time, (F - S) : (M - S) = 11 : 10. Step 4: This gives the equation (F - S) / (M - S) = 11 / 10. Step 5: In the future, when the son is twice his present age, his age will be 2S. To reach that age from S, a time interval of S years must pass. Step 6: After S years, the father age will be F + S and the mother age will be M + S. Step 7: At that time, (F + S) : (M + S) = 19 : 18, so (F + S) / (M + S) = 19 / 18. Step 8: From the first ratio, write F - S = 11k and M - S = 10k for some positive constant k. Step 9: From the second ratio, write F + S = 19t and M + S = 18t for some constant t. Step 10: Solving these equations gives F = 15t and M = 14t, which means the present age ratio F : M is 15 : 14.

Verification / Alternative check:
To verify, assume t = 1 for simplicity. Then F = 15 and M = 14. The son present age from the equations would be S = 4. At birth, the father and mother would be 11 and 10 respectively, giving the required ratio 11 : 10. After 4 years, when the son is 8 years old, the father and mother would be 19 and 18 respectively, giving the second ratio 19 : 18. Both conditions match perfectly, so the ratio 15 : 14 is correct.

Why Other Options Are Wrong:
Option B (14 : 13) cannot satisfy both sets of ratio conditions when tested in a similar way, so it is invalid.
Option C (16 : 15) also fails when substituted into the past and future relationships, leading to contradictions.
Option D (17 : 16) similarly does not produce the specified ratios of 11 : 10 and 19 : 18 at the required times.

Common Pitfalls:
A frequent error is to treat ratios as if they directly compare present ages without accounting for the changes over time. Another pitfall is to assume a fixed difference that does not respect the given ratios. The safe method is to introduce variables and constants for ratios, write equations carefully and then solve systematically. Working with symbolic constants like k and t helps keep the algebra manageable.

Final Answer:
The ratio of the present ages of the father and the mother is 15 : 14.