Question

The mean of $11$ observations is $50$. If the mean of the first six observations is $49$ and that of the last six observations is $52$, which option is the sixth observation?
A. $55$
B. $56$
C. $57$
D. $58$

Answer 1

We will be using the formula of mean which is the average of the numbers as shown below:
${\text{Mean}} = \dfrac{{{\text{sum of the terms}}}}{{{\text{number of terms}}}}$
By using this formula, we will be calculating the sum of the terms.

Complete step-by-step solution:
Step 1: Total number of observations are
$11$ and the mean is
$50$. By using the formula of mean we get:
$\Rightarrow {\text{Mean}} = \dfrac{{{\text{sum of the terms}}}}{{11}}$
By substituting the value of mean in the above expression we get:
$\Rightarrow 50 = \dfrac{{{\text{sum of the terms}}}}{{11}}$
By taking $11$ into the LHS side of the above expression, we get:
$\Rightarrow 50 \times 11 = {\text{sum of the terms}}$
By doing the final multiplication in the above expression we get:
$\Rightarrow {\text{sum of the terms = 550}}$ …………………………. (1)
Step 2: The mean of the first six observations is $49$. By using the formula of mean we get:
$\Rightarrow {\text{Mean}} = \dfrac{{{\text{sum of the terms}}}}{6}$
By substituting the value of mean in the above expression we get:
$\Rightarrow 49 = \dfrac{{{\text{sum of the terms}}}}{6}$
By taking $6$ into the LHS side of the above expression, we get:
$\Rightarrow 49 \times 6 = {\text{sum of the terms}}$
By doing the final multiplication in the above expression we get:
$\Rightarrow {\text{sum of the terms = 294}}$ …………………………. (2)
Step 3: The mean of the first six observation is
$52$. By using the formula of mean we get:
$\Rightarrow {\text{Mean}} = \dfrac{{{\text{sum of the terms}}}}{6}$
By substituting the value of mean in the above expression we get:
$\Rightarrow 52 = \dfrac{{{\text{sum of the terms}}}}{6}$
By taking $6$ into the LHS side of the above expression, we get:
$\Rightarrow 52 \times 6 = {\text{sum of the terms}}$
By doing the final multiplication in the above expression we get:
$\Rightarrow {\text{sum of the terms = 312}}$ …………………………. (3)
Step 4: Now the sixth observation will be equals to the subtraction of summation of the first and last six observation from the total sum as shown below:
$\Rightarrow {\text{Sixth observation = }}\left( {{\text{sum of first six + sum of last six}}} \right) - {\text{sum of total observation}}$
By substituting the values from expression (1), (2), and (3) in the above one we get:
$\Rightarrow {\text{Sixth observation = }}\left( {{\text{294 + 312}}} \right) - 550$
Solving the brackets first by doing addition we get:
$\Rightarrow {\text{Sixth observation = }}\left( {606} \right) - 550$
By doing the final subtraction in the above expression we get:
$\Rightarrow {\text{Sixth observation = 56}}$

Option B is correct.

Note: Students should remember in solving these types of questions that for finding any middle term in the series when the mean is given for the rest then we need to subtract the sum of the series from the total of that series.