Question

The mean of 15 observations is 21.2. If the mean of the first seven observations is 18.5, and that of the last seven observations is 22.5, then the value of the $8^{th}$ observation is

• A. 24
• B. 29
• C. 30
• D. 31

Answer 1

The correct option is D 31

Mean of observations = $\frac{\text{Sum of observations}}{\text{Total number of observations}}$
$\Rightarrow$ Sum of observations = Mean $\times$ Total number of observations
$\Rightarrow$ Sum of 15 observations = 21.2 $\times$ 15 ..(i)
There are 15 observations in total and the mean of first seven observations is 18.5 and last seven observations is 22.5.
$\therefore 18.5=\frac{\text{Sum of first 7 observations}}{7}$
$\Rightarrow$ Sum of first 7 observations = 18.5 $\times$ 7 …..(ii)
Also, $22.5=\frac{\text{Sum of last 7 observations}}{7}$
$\Rightarrow$ Sum of last 7 observations = 22.5 $\times$ 7 …..(iii)
Now, Sum of total 15 observations = Sum of first 7 observations + $8^{th}$ observation + Sum of last 7 observations
$\Rightarrow$ $8^{th}$ observation = Sum of total observations – (Sum of first seven and last seven observations)
$=21.2\times 15–(18.5\times 7+22.5\times 7)$ [From (i), (ii) and (iii)]
$=318–(129.5+157.5)$
$=318–287$
$=31$
Hence, the correct answer is option (d).