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.

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Solution

## We have, The mean of 35 observations is 20. Let x be the sum of 35 observations $⇒$ mean = $\frac{\mathrm{sum}\mathrm{of}\mathrm{observations}}{\mathrm{number}\mathrm{of}\mathrm{observations}}$ $⇒20=\frac{x}{35}\phantom{\rule{0ex}{0ex}}⇒x=700$ Let y be the sum of first 18 observations. Mean of first 18 observations is 15. $\frac{y}{18}=15\phantom{\rule{0ex}{0ex}}⇒y=270$ And z be the sum of last 18 observations Mean of last 18 observations is 25. $\frac{z}{18}=25\phantom{\rule{0ex}{0ex}}⇒z=450$ let the 18th term be a. 18th term is common in first 18 and last 18 observations $x+a=y+z\phantom{\rule{0ex}{0ex}}700+a=270+450\phantom{\rule{0ex}{0ex}}a=720-700\phantom{\rule{0ex}{0ex}}a=20$ Hence, the 18th observation is 20.

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