Question

The mean of 35 observations is 20, out of which mean of first 18 observations is 15 and mean of last 18 observation is 25. Find the 18th observation.

Answer 1

We have,

The mean of 35 observations is 20.
Let x be the sum of 35 observations

$\Rightarrow$ mean = $\frac{\mathrm{sum}\mathrm{of}\mathrm{observations}}{\mathrm{number}\mathrm{of}\mathrm{observations}}$
$$\begin{gathered} \Rightarrow 20=\frac{x}{35} \\ \Rightarrow x=700 \end{gathered}$$

Let y be the sum of first 18 observations.
Mean of first 18 observations is 15.
$$\begin{gathered} \frac{y}{18}=15 \\ \Rightarrow y=270 \end{gathered}$$
And z be the sum of last 18 observations
Mean of last 18 observations is 25.
$$\begin{gathered} \frac{z}{18}=25 \\ \Rightarrow z=450 \end{gathered}$$
let the 18th term be a.
18th term is common in first 18 and last 18 observations
$$\begin{gathered} x+a=y+z \\ 700+a=270+450 \\ a=720-700 \\ a=20 \end{gathered}$$
Hence, the 18th observation is 20.