Question

The mean of 19 observations is 36. If the mean of first 10 observations is 35 and that of the last 10 observations is 37, then the ${10}^{th}$ observation is

• A. 36
• B. 37
• C. 35
• D. 34

Answer 1

The correct option is A 36

Let the 19 observations be $x_1,x_2,x_3,....,x_{19}.$
Mean =$\frac{\text{Sum of observations}}{\text{Number of observations}}$
We have,$\frac{x_1+x_2+....+x_{19}}{19}=36$
$\Rightarrow x_1+x_2+x_3+....+x_{19}=36\times 19=684$ .......(i)
and $\frac{x_1+x_2+....+x_{10}}{10}=35$
$\Rightarrow x_1+x_2+x_3+....+x_{10}=35\times 10=350$ .......(ii)
and $\frac{x_{10}+x_{11}+....+x_{19}}{10}=37$
$\Rightarrow x_{10}+x_{11}+....+x_{19}=37\times 10=370$ .......(iii)
Adding (ii) in (iii), we get
$x_1+x_2+x_3+....+x_{10}+x_{11}+....+x_{19}=350+370$
$\Rightarrow (x_1+x_2+....,x_{19})+x_{10}=720$
Putting (i) in (iv), we get
$684+x_{10}=720$
$\Rightarrow x_{10}=720-684$
$\Rightarrow x_{10}=36$
Therefore, the ${10}^{th}$ observation is 36.
Hence, the correct answer is option (a).