The present ages of Sunil and Anil are in the ratio 8 : 7. After 6 years, the ratio of their ages will be 19 : 17. What is the difference between their present ages in years?

Introduction / Context:
This question is another example of a two time ratio problem on ages. It provides the current ratio of ages and the ratio after a certain number of years. From this information you have to deduce the actual ages and then compute their difference. Problems like this test both algebraic manipulation and understanding of how adding the same number to both terms of a ratio affects that ratio.

Given Data / Assumptions:

• Present ages of Sunil and Anil are in the ratio 8 : 7.
• After 6 years, their ages will be in the ratio 19 : 17.
• We are required to find the difference between their present ages.
• Both persons age normally by 6 years over the given period.

Concept / Approach:
The standard approach is to represent the present ages as multiples of a common variable, using the given ratio. Then we write expressions for their ages after 6 years and equate the resulting ratio to the second one. This gives a linear equation in the variable. Solving it yields the variable, from which we can compute the actual ages and their difference. This method relies on the fact that ratios provide relative information which becomes absolute once one more equation is known.

Step-by-Step Solution:
Step 1: Let the present ages of Sunil and Anil be 8k and 7k years respectively. Step 2: After 6 years, their ages will be 8k + 6 and 7k + 6 years. Step 3: The future ratio condition says (8k + 6) : (7k + 6) = 19 : 17. Step 4: Form the equation (8k + 6) / (7k + 6) = 19 / 17 and solve for k. Step 5: Solving gives k = 4, so the present ages are 32 years for Sunil and 28 years for Anil. Step 6: The difference between their present ages is 32 − 28 = 4 years.

Verification / Alternative check:
Check the ratios with the found ages. The present ratio 32 : 28 simplifies to 8 : 7, which matches the given ratio. After 6 years, the ages will be 38 and 34, and the ratio 38 : 34 reduces to 19 : 17, which matches the future ratio. Since both ratios come out correctly, the difference of 4 years is confirmed as accurate and the solution is consistent with all conditions.

Why Other Options Are Wrong:
Option 10 years, 6 years, and 8 years: These larger differences would generate present ages that do not fit both ratios at the same time. Testing these differences with suitable pairs fails either the present or future ratio condition.
Option 2 years: This difference is too small; if you try to construct ages with a 2 year gap and ratio 8 : 7, the arithmetic does not work correctly.

Common Pitfalls:
A common error is to forget to add 6 years to both ages in the future ratio or to write the equation with the ratios reversed. Some learners also try to guess ages instead of systematically solving the algebra, which can be time consuming and prone to mistakes. Writing the ratio equation clearly, cross multiplying carefully, and checking both ratios at the end ensures a reliable solution.

Final Answer:
The difference between the present ages of Sunil and Anil is 4 years.