Sumitra remembers that her mother birthday is after 13th February but before 16th February, while her brother remembers that the birthday is after 14th February but before 17th February. On which exact date will their mother birthday be celebrated?

Introduction / Context:
This logical reasoning question asks you to determine an exact date based on two overlapping pieces of partial information. Two siblings remember their mother birthday only within certain ranges of dates. The goal is to combine these ranges to find the single date that satisfies both memories. It is a simple but important example of using intersections of intervals to find a precise result.

Given Data / Assumptions:

• Sumitra remembers that the birthday is after 13th February but before 16th February.
• Her brother remembers that the birthday is after 14th February but before 17th February.
• Dates are in February of the same year.
• The birthday is on exactly one specific day, not spread over several days.
• The wording “after” and “before” means the endpoints themselves are excluded.

Concept / Approach:
Each child is giving a date range. Sumitra narrows the birthday down to a small set of possible dates, and her brother narrows it down to another set. The correct date must be one that lies inside both ranges simultaneously. Mathematically, it is the intersection of the two sets of possible dates. So we list the dates allowed by each memory and then find the common date.

Step-by-Step Solution:
Step 1: Interpret Sumitra statement. The birthday is after 13th February but before 16th February. That means it can be on 14th or 15th February, because 14 and 15 are between 13 and 16, not including the endpoints.Step 2: So according to Sumitra, the possible dates are {14th, 15th} February.Step 3: Now interpret the brother statement. The birthday is after 14th February but before 17th February. That means it can be on 15th or 16th February.Step 4: So according to the brother, the possible dates are {15th, 16th} February.Step 5: The actual birthday must belong to both sets of possible dates. Therefore we take the intersection of the sets {14th, 15th} and {15th, 16th}.Step 6: The only date common to both sets is 15th February.Step 7: Hence the birthday will be celebrated on 15th February.

Verification / Alternative check:
We can also visualise the dates on a number line. Mark 13, 14, 15, 16 and 17 as points. Sumitra valid region is between 13 and 16, excluding both, which covers 14 and 15 only. The brother valid region is between 14 and 17, excluding both, which covers 15 and 16 only. The overlap of these two highlighted segments is the single point at 15. This confirms that all given information points to the same date.

Why Other Options Are Wrong:
13th February and 14th February are not possible because Sumitra said the birthday is after 13th and the brother said it is after 14th, so neither 13 nor 14 fits the second statement. 17th February is ruled out because both children said “before” 16th or 17th, excluding those end dates. Only 15th February lies inside both specified ranges.

Common Pitfalls:
A typical mistake is to include the endpoints when the phrases clearly say “after” and “before”. Another error is to take the union of the ranges instead of the intersection, which would enlarge rather than narrow the set of possible dates. Always pay close attention to whether endpoints are included or excluded and whether you need common elements or just any element from either range.

Final Answer:
Sumitra mother birthday will be celebrated on 15th February.