In Arun's opinion, his weight is greater than $65$ kg but less than $72$ kg. His brother does not agree with Arun and he thinks that Arun's weight is greater than $60$ kg but less than $70$ kg. His mother's view is that his weight cannot be greater than $68$ kg. If all of them are correct in their estimation, what is the average of different probable weights of Arun?

### Concept & Logic This is an intersecting sets (inequalities) problem. To satisfy multiple conditions simultaneously, we must find the overlapping range of all given estimations. "Probable weights" implies we are looking for discrete, integer values within this final valid range. ### Step-by-Step Solution * **Given Conditions:** Arun's estimation: $65 < \text{Weight} < 72$ Brother's estimation: $60 < \text{Weight} < 70$ Mother's estimation: $\text{Weight} \le 68$ (cannot be greater than 68) * **Deduction:** Step 1: Combine the lower bounds (greater than). Arun says $> 65$, Brother says $> 60$. For both to be correct, the weight must be strictly greater than $65$. Step 2: Combine the upper bounds (less than or equal to). Arun says $< 72$, Brother says $< 70$, Mother says $\le 68$. For all to be correct, the most restrictive upper limit applies. It must be less than or equal to $68$. Step 3: Define the final valid range. $65 < \text{Weight} \le 68$ Step 4: Identify the probable integer weights. The integers strictly greater than $65$ and less than or equal to $68$ are $66$, $67$, and $68$. * **Calculation:** Average of these probable weights: $\text{Average} = \frac{66 + 67 + 68}{3}$ Since they form an arithmetic progression, the average is the middle number. $\text{Average} = 67$ kg. ### Exam Strategy & Shortcut **Number Line Visualization:** Quickly sketch or mentally picture a number line. Mark the strictest lower bound ($65$ from Arun) and the strictest upper bound ($68$ from Mother). The numbers trapped inside are $66$, $67$, and $68$. The middle number of a consecutive sequence is always its average, giving $67$ instantly without doing actual addition and division. ### Common Pitfall Misinterpreting the phrase "cannot be greater than $68$ kg". Students often take this to mean strictly less than $68$ ($< 68$), which would leave only $66$ and $67$ as probable weights, resulting in an incorrect average of $66.5$. "Cannot be greater than" mathematically translates to "less than or equal to" ($\le$). ### Final Answer **Therefore, the correct answer is 67 kg.**