The present ages of R and S are in the ratio 6 : 7. Twelve years ago, the ratio of their ages was 9 : 11. What is the present age of R (in years)?

Introduction / Context:
This is a typical ages and ratio problem. It involves the present ratio of ages of two people and their ratio of ages a certain number of years in the past. The problem checks understanding of how ages change over time and how to set up equations based on ratios to determine actual ages.

Given Data / Assumptions:

• Present age ratio of R and S is 6 : 7.• Twelve years ago, the ratio of their ages was 9 : 11.• Ages increase uniformly over time by the same number of years for each person.• We must find R's present age in years.

Concept / Approach:
We represent the present ages using a common multiplier. Let R's present age be 6k and S's present age be 7k for some positive integer k. Twelve years ago, their ages would have been 6k − 12 and 7k − 12. At that time, the ratio of their ages was 9 : 11. This gives the equation (6k − 12) / (7k − 12) = 9 / 11. Solving this equation for k allows us to find the actual present ages, particularly R's age.

Step-by-Step Solution:
Step 1: Let present age of R be 6k and present age of S be 7k.Step 2: Twelve years ago, R's age would have been 6k − 12 and S's age would have been 7k − 12.Step 3: We are told that twelve years ago, their ages were in the ratio 9 : 11, so (6k − 12) / (7k − 12) = 9 / 11.Step 4: Cross multiply to eliminate the fraction: 11 * (6k − 12) = 9 * (7k − 12).Step 5: Expand both sides: 66k − 132 = 63k − 108.Step 6: Bring k terms to one side and constants to the other: 66k − 63k = −108 + 132.Step 7: This simplifies to 3k = 24.Step 8: So k = 24 / 3 = 8.Step 9: Therefore, R's present age = 6k = 6 * 8 = 48 years.Step 10: S's present age could also be found as 7k = 7 * 8 = 56 years if needed.

Verification / Alternative check:
We can verify by checking the ratio 12 years ago. R's age 12 years ago = 48 − 12 = 36 years. S's age 12 years ago = 56 − 12 = 44 years. The ratio 36 : 44 simplifies by dividing both numbers by 4 to 9 : 11, which matches the condition given in the problem. The present ratio 48 : 56 simplifies by dividing by 8 to 6 : 7, which matches the given present ratio. These checks confirm that the solution is consistent.

Why Other Options Are Wrong:
• 56: This is the present age of S, not R.• 36: This is R's age twelve years ago, not the present age.• 44: This does not correspond to either the present or past age of R in the correct ratio structure.

Common Pitfalls:
Some learners confuse which person corresponds to which part of the ratio and may assign 7k to R and 6k to S, which leads to an incorrect equation. Others forget to subtract 12 years from both ages when forming the past ratio, or they set up the ratio in reverse order. Algebraic mistakes, such as expansion errors or sign errors when moving terms across the equation, can also obstruct finding the correct value of k. Careful mapping of the ratios and neat algebraic work help avoid these issues.

Final Answer:
The present age of R is 48 years.