The ratio of the present ages of R and S is 11 : 17. Eleven years ago, the ratio of their ages was 11 : 20. What is R's present age (in years)?

Introduction / Context:
This question deals with present and past ratios of the ages of two people, R and S. We know their current age ratio and their age ratio eleven years ago. Using these two relations, we must determine R's present age. This is a classic linear equation and ratio problem from aptitude tests on ages.

Given Data / Assumptions:

- Present age ratio of R : S = 11 : 17.
- Eleven years ago, the ratio of their ages was 11 : 20.
- All ages are positive and measured in years.

Concept / Approach:
We represent the present ages of R and S as multiples of a common variable using the present ratio. Then we subtract 11 years from each to obtain their ages eleven years ago and form the second ratio equation. Solving the resulting algebraic equation gives the value of the variable and hence R's present age.

Step-by-Step Solution:
Step 1: Let the present ages of R and S be 11x years and 17x years respectively. Step 2: Eleven years ago, R's age was 11x − 11 years and S's age was 17x − 11 years. Step 3: The ratio eleven years ago is given as (11x − 11) : (17x − 11) = 11 : 20. Step 4: Form the equation (11x − 11) / (17x − 11) = 11 / 20. Step 5: Cross-multiply: 20(11x − 11) = 11(17x − 11). Step 6: Expand: 220x − 220 = 187x − 121. Step 7: Rearrange to get 220x − 187x = 220 − 121 ⇒ 33x = 99 ⇒ x = 3. Step 8: R's present age = 11x = 11 × 3 = 33 years.

Verification / Alternative check:
With x = 3, S's present age is 17 × 3 = 51 years. Eleven years ago, R was 33 − 11 = 22 years and S was 51 − 11 = 40 years. The ratio 22 : 40 simplifies to 11 : 20, which matches the given past ratio. The present ratio is 33 : 51, which simplifies to 11 : 17 when divided by 3. Both conditions are satisfied.

Why Other Options Are Wrong:
If R were 22, 40, 51 or 55 years old at present, the resulting ages and ratios eleven years ago would fail to match both the given present ratio 11 : 17 and the past ratio 11 : 20. Only 33 years leads to consistent ages for both time points.

Common Pitfalls:
A frequent mistake is to treat the ratio numbers directly as ages or to forget to subtract 11 years from both ages when forming the past ratio. Another error involves incorrect cross-multiplication or simplification of equations. Always maintain the structure: present ages as multiples of x, subtract the time shift, then form and solve the ratio equation.

Final Answer:
R's present age is 33 years.