The average of 11 observations is 60. If the average of first five observations is 58 and that of the last five is 56, then the sixth observation is

Options

A. a) 90
B. b) 110
C. c) 85
D. d) 100

Answer

To solve the problem step by step, we will follow these calculations:

**Step 1: Calculate the total sum of all 11 observations.**
- The average of 11 observations is given as 60.
- Total sum of observations = Average × Number of observations = 60 × 11 = 660.

**Step 2: Calculate the total sum of the first five observations.**
- The average of the first five observations is given as 58.
- Total sum of the first five observations = Average × Number of observations = 58 × 5 = 290.

**Step 3: Calculate the total sum of the last five observations.**
- The average of the last five observations is given as 56.
- Total sum of the last five observations = Average × Number of observations = 56 × 5 = 280.

**Step 4: Set up the equation to find the sixth observation.**
- Let the sixth observation be represented as \( x \).
- The total sum of all observations can be expressed as:
\[ \text{Sum of first five} + \text{Sixth observation} + \text{Sum of last five} = \text{Total sum} \]
Thus, we have:
\[ 290 + x + 280 = 660 \]

**Step 5: Solve for the sixth observation \( x \).**
- Combine the sums of the first five and last five observations:
\[ 290 + 280 = 570 \]
- Substitute this back into the equation:
\[ 570 + x = 660 \]
- Now, isolate \( x \):
\[ x = 660 - 570 \]
\[ x = 90 \]

**Final Answer:**
The sixth observation is 90.

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