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?

Options

A. 67 kg
B. 68 kg
C. 69 kg
D. None of these

Answer

To find the average of the different probable weights of Arun based on the opinions of Arun, his brother, and his mother, we can follow these steps:

### Step 1: Establish the weight ranges based on each person's opinion.
- **Arun's opinion**: His weight (X) is greater than 65 kg but less than 72 kg. This can be expressed as:
\[ 65 < X < 72 \]

- **Brother's opinion**: He thinks Arun's weight is greater than 60 kg but less than 70 kg:
\[ 60 < X < 70 \]

- **Mother's opinion**: She believes Arun's weight cannot be greater than 68 kg:
\[ X \leq 68 \]

### Step 2: Combine the conditions to find the overlapping range.
To find the valid range for Arun's weight, we need to combine all three conditions:
1. From Arun: \( X > 65 \) and \( X < 72 \)
2. From his brother: \( X > 60 \) and \( X < 70 \)
3. From his mother: \( X \leq 68 \)

The overlapping range from these conditions is:
- The lower limit is 65 (from Arun's opinion).
- The upper limit is 68 (from his mother's opinion).

Thus, we have:
\[ 65 < X \leq 68 \]

### Step 3: Identify the possible integer weights.
The possible integer weights that satisfy the condition \( 65 < X \leq 68 \) are:
- 66 kg
- 67 kg
- 68 kg

### Step 4: Calculate the average of these weights.
To find the average of the weights 66 kg, 67 kg, and 68 kg, we use the formula for the average:
\[ \text{Average} = \frac{\text{Sum of weights}}{\text{Number of weights}} = \frac{66 + 67 + 68}{3} \]

Calculating the sum:
\[ 66 + 67 + 68 = 201 \]

Now, divide by the number of weights (which is 3):
\[ \text{Average} = \frac{201}{3} = 67 \]

### Final Answer:
The average of the different probable weights of Arun is **67 kg**.

---