Ram is 20 years younger than Shyam. Five years ago, the ratio of their ages was 3 : 5. What is the sum of their present ages?

Introduction / Context:
This is a classic age-based word problem where we are given the age difference between two people and a ratio of their ages at a previous time. We must use this information to compute their current ages and then find the sum of those ages.

Given Data / Assumptions:

- Ram is 20 years younger than Shyam at present.
- Five years ago, the ratio of their ages was 3 : 5 (Ram : Shyam).
- Ages of both increase by 5 years from five years ago to the present.
- We are asked for the sum of their present ages.

Concept / Approach:
We set up algebraic expressions for their present ages using a variable for Shyam's age. Next we form an equation from the ratio given for their ages five years ago. Solving this equation gives the present ages of both Ram and Shyam, from which we compute their sum. This problem uses linear equations and the idea that age differences remain constant over time.

Step-by-Step Solution:
Step 1: Let Shyam's present age be S years. Then Ram's present age is S − 20 years. Step 2: Five years ago, Shyam's age was S − 5 and Ram's age was (S − 20) − 5 = S − 25. Step 3: The ratio five years ago was (Ram : Shyam) = 3 : 5, so (S − 25) / (S − 5) = 3 / 5. Step 4: Cross-multiply to get 5(S − 25) = 3(S − 5). Step 5: Expand: 5S − 125 = 3S − 15. Step 6: Rearrange to get 2S = 110, so S = 55 years. Step 7: Ram's present age = S − 20 = 55 − 20 = 35 years. Step 8: Sum of their present ages = 55 + 35 = 90 years.

Verification / Alternative check:
Check the ages five years ago: Ram = 35 − 5 = 30 years and Shyam = 55 − 5 = 50 years. The ratio 30 : 50 simplifies to 3 : 5, which matches the given ratio. The sum of their current ages is indeed 90 years, confirming our solution.

Why Other Options Are Wrong:
75 years, 80 years, 85 years, and 95 years are all sums that do not arise from any integer solution that fits the ratio 3 : 5 five years ago and the 20-year age difference. Only 90 years satisfies both the ratio condition and the fixed difference in their ages.

Common Pitfalls:
Students sometimes mix up which age is older or misapply the ratio at the wrong time (present instead of five years ago). Another frequent mistake is to ignore the constant age difference information, which is crucial in forming the correct equation. Carefully aligning the time frame and the given ratio avoids these issues.

Final Answer:
The sum of Ram's and Shyam's present ages is 90 years.