The average of 11 observations is 60. If the average of first six observations is 59 and that of last six observations is 62, then the value of sixth observation, is :

Question

Accepted Answer

To solve the problem and confirm why Option 3 is the correct answer, we need to analyze the given information and perform some calculations.

### Given Information:
1. The average of 11 observations is 60.
2. The average of the first six observations is 59.
3. The average of the last six observations is 62.

### Calculations:

1. **Total Sum of 11 Observations:**
   - Since the average of 11 observations is 60, the total sum of these observations is:
     $$
     11 \times 60 = 660
     $$

2. **Total Sum of First Six Observations:**
   - The average of the first six observations is 59, so their total sum is:
     $$
     6 \times 59 = 354
     $$

3. **Total Sum of Last Six Observations:**
   - The average of the last six observations is 62, so their total sum is:
     $$
     6 \times 62 = 372
     $$

4. **Finding the Sixth Observation:**
   - The sixth observation is included in both the first six and the last six observations. To find its value, we can use the equation:
     $$
     \text{Sum of first six observations} + \text{Sum of last six observations} - \text{Total sum of all observations} = \text{Sixth observation} 
     $$
   - Plugging in the values:
     $$
     354 + 372 - 660 = 66
     $$

Thus, the value of the sixth observation is 66, which corresponds to Option 3.

### Explanation of Why Other Options are Incorrect:

- **Option 1 (63):** If the sixth observation were 63, the calculations would not balance as shown above. The sum of the first six and last six observations minus the total would not equal 63.
  
- **Option 2 (64):** Similarly, if the sixth observation were 64, the equation would not hold true. The calculated value must be 66 to satisfy the given conditions.

- **Option 4 (68):** If the sixth observation were 68, the sum of the first six and last six observations minus the total would not equal 68. The correct calculation shows it must be 66.

Therefore, based on the calculations and logical reasoning, Option 3 is indeed the correct answer.