Question

11 natural numbers are listed sequentially. The mean of the 11 numbers is 50. If the average of first 6 numbers is 49 and that of the last 6 numbers is 52, then the sixth number is

• A. 50
• B. 58
• C. 56
• D. 51

Hint:

Whenever a set of numbers overlaps, the sum of the overlapping groups minus the total sum will directly yield the value of the overlapping element.
You can also use deviations from the mean to solve this mentally.
The first \( 6 \) numbers have a deviation of \( -1 \) each (total \( -6 \)).
The last \( 6 \) numbers have a deviation of \( +2 \) each (total \( +12 \)).
The net deviation is \( +6 \), which means the overlapping term is \( 50 + 6 = 56 \).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given a sequence of eleven natural numbers.
The total mean is given, as well as the sub-means for overlapping groups of the first six and the last six numbers.
The sixth number appears in both groups, causing it to be counted twice in their combined sum.


Step 2: Key Formula or Approach:
The fundamental formula for calculating the arithmetic mean is:
\[ \text{Mean} = \frac{\text{Sum of Observations}}{\text{Number of Observations}} \]
From this, we can derive the sum of the observations:
\[ \text{Sum} = \text{Mean} \times \text{Number of Observations} \]
Let the numbers be sequentially denoted as \( x_1, x_2, \dots, x_{11} \).
The sum of the first six numbers is \( S_1 = x_1 + x_2 + x_3 + x_4 + x_5 + x_6 \).
The sum of the last six numbers is \( S_2 = x_6 + x_7 + x_8 + x_9 + x_{10} + x_{11} \).
The total sum of all eleven numbers is \( S = x_1 + x_2 + \dots + x_{11} \).
Observe that \( S_1 + S_2 = (x_1 + \dots + x_{11}) + x_6 = S + x_6 \).
Thus, the sixth number can be found as \( x_6 = S_1 + S_2 - S \).


Step 3: Detailed Explanation:

• First, we calculate the sum of all eleven natural numbers using the given mean.

• Given that the mean of the \( 11 \) numbers is \( 50 \).

• Sum of \( 11 \) numbers is:
\[ S = 11 \times 50 = 550 \]

• Next, we find the sum of the first six numbers.

• Given that the average of the first \( 6 \) numbers is \( 49 \).

• Sum of first \( 6 \) numbers is:
\[ S_1 = 6 \times 49 = 294 \]

• Then, we find the sum of the last six numbers.

• Given that the average of the last \( 6 \) numbers is \( 52 \).

• Sum of last \( 6 \) numbers is:
\[ S_2 = 6 \times 52 = 312 \]

• Now, we sum the two overlapping sets.

• This combined sum is:
\[ S_1 + S_2 = 294 + 312 = 606 \]

• According to our key relation, this combined sum is equal to the total sum of all eleven numbers plus the sixth number itself.

• Therefore, we substitute the values into the equation:
\[ 606 = 550 + x_6 \]

• Solving for the sixth number \( x_6 \):
\[ x_6 = 606 - 550 = 56 \]



Step 4: Final Answer:
The sixth number in the sequence is \( 56 \), which corresponds to Option (C).