The ratio of the present ages of Ranjini and Shahid is 5 : 4. After 13 years, the ratio of their ages will be 6 : 5. What is Ranjini present age in years?

Introduction / Context:
This is a typical two time ratio problem on ages involving Ranjini and Shahid. The present ratio of their ages is given, and another ratio is given for their ages after a fixed period of 13 years. Your task is to find Ranjini present age. Such problems test your skill in setting up algebraic expressions for ages at different times and solving the resulting equations.

Given Data / Assumptions:

• The present age ratio of Ranjini to Shahid is 5 : 4.
• After 13 years, their age ratio will be 6 : 5.
• We have to determine Ranjini present age.
• Both ages increase linearly by 13 years over the period.

Concept / Approach:
Let the present ages be 5k and 4k for Ranjini and Shahid respectively. After 13 years, their ages become 5k + 13 and 4k + 13. The future ratio condition gives a fraction equation involving k. Solving this equation yields the value of k and hence the actual ages. The method uses basic algebra and the idea that adding a fixed number to both terms of a ratio changes the ratio in a predictable way.

Step-by-Step Solution:
Step 1: Let Ranjini present age be 5k years and Shahid present age be 4k years. Step 2: After 13 years, their ages will be 5k + 13 and 4k + 13 years. Step 3: According to the question, (5k + 13) : (4k + 13) = 6 : 5. Step 4: Convert this into the equation (5k + 13) / (4k + 13) = 6 / 5 and solve for k. Step 5: Solving the equation gives k = 13. Thus, Ranjini present age is 5k = 5 × 13 = 65 years. Step 6: Therefore, the required present age of Ranjini is 65 years.

Verification / Alternative check:
Check using Ranjini age 65 and Shahid age 52 (since 4 × 13 = 52). Their present ratio is 65 : 52, which simplifies to 5 : 4 by dividing both numbers by 13. After 13 years, their ages will be 78 and 65 respectively. The ratio 78 : 65 simplifies to 6 : 5 by dividing by 13 again. Both ratio conditions are satisfied, confirming that 65 is Ranjini correct present age.

Why Other Options Are Wrong:
Option 52 and 60: These do not maintain both the present and future ratios when you back calculate Shahid age and check the conditions.
Option 32 and 40: These values are too small and lead to future ages and ratios that do not match the given 6 : 5 relationship after 13 years.

Common Pitfalls:
Learners sometimes swap the roles of Ranjini and Shahid when assigning the ratio parts, which leads to incorrect equations. Another common mistake is to forget to add 13 years to both ages before forming the second ratio. Writing the ratio equation carefully and double checking which term corresponds to which person helps prevent these errors.

Final Answer:
Ranjini present age is 65 years.