Standing on a platform, Rajan says that Varanasi is more than 20 km but less than 25 km from there. Kumar says that it is more than 18 km but less than 22 km from there. If both of them are correct, what is the true distance of Varanasi from the platform?

Introduction / Context:
This question is a classic example of reasoning with inequalities and overlapping ranges. Two people give different but partly overlapping distance estimates for Varanasi from the same platform. You are told that both of them are correct, which means the actual distance must satisfy both sets of conditions at the same time. The task is to find which option fits within both ranges.

Given Data / Assumptions:
- Rajan states that the distance is more than 20 km and less than 25 km. - This can be written as 20 km < distance < 25 km. - Kumar states that the distance is more than 18 km and less than 22 km. - This can be written as 18 km < distance < 22 km. - Both statements are assumed to be correct simultaneously, so the distance must lie in the intersection of these two ranges.

Concept / Approach:
To satisfy both inequalities at the same time, the distance must lie in the overlapping portion of the two intervals. This means the distance must be greater than the larger of the two lower limits and less than the smaller of the two upper limits. So we find max(20,18) and min(25,22), then check which of the given options lies strictly between those values. Because the limits are given as more than and less than, the endpoints 20 and 22 themselves are excluded from the possible values.

Step-by-Step Solution:
Step 1: Write Rajan's inequality: 20 < distance < 25. Step 2: Write Kumar's inequality: 18 < distance < 22. Step 3: For both to be true together, the distance must satisfy both inequalities. Step 4: The distance must be greater than 20 km, because that is the higher of the two lower bounds (20 km and 18 km). Step 5: The distance must be less than 22 km, because that is the lower of the two upper bounds (25 km and 22 km). Step 6: Therefore, the combined condition is 20 km < distance < 22 km. Step 7: Check the answer options: 20 km is equal to the lower bound and therefore not strictly greater; 22 km is equal to the upper bound and not strictly less. Step 8: 19 km is less than 20 km and fails Rajan's condition. Step 9: 21 km is greater than 20 km and less than 22 km, so it satisfies both Rajan's and Kumar's statements. Step 10: Thus, the only possible true distance consistent with both is 21 km.

Verification / Alternative check:
You can visualise both ranges on a number line. Mark an open interval from just above 20 to just below 25 for Rajan, and another from just above 18 to just below 22 for Kumar. The overlapping region is from just above 20 to just below 22. Among the integers in the options, only 21 lies inside this region. All other listed distances stay outside at least one of the two intervals, which confirms that 21 km is the only value that makes both friends correct.

Why Other Options Are Wrong:
20 km: Rajan says the distance is more than 20 km, so 20 km itself cannot be correct. 22 km: Kumar says the distance is less than 22 km, so 22 km itself is excluded. 19 km: This contradicts Rajan's statement that the distance is more than 20 km.

Common Pitfalls:
Students sometimes overlook the strict wording more than and less than and mistakenly include the boundary values. Another common error is to average the given ranges or the two midpoints instead of calculating the intersection. In interval questions, always pay attention to whether the inequalities are strict or allow equality, and focus on where the two conditions overlap rather than simply mixing or averaging numbers.

Final Answer:
The distance of Varanasi from the platform must be 21 km to satisfy both statements.