The mean of 9 observations is 26 if the mean first 5 observation is 28 and that of last five observations is 24 find the fifth observation
Answers
Question:
The mean of nine observations is 26 if the mean first five observation is 28 and that of last five observations is 24. Find the fifth observation.
Answer:
Fifth observation is 26.
Note:
• Mean or average is given as
Sum of observations/total no. of observations
Solution:
Let the nine observations be ;
a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 , a9 .
According to the question;
The mean of nine observations is 26.
=> (a1+a2+a3+a4+a5+a6+a7+a8+a9)/9 = 26
=> a1+a2+a3+a4+a5+a6+a7+a8+a9 = 26•9
=> a1+a2+a3+a4+a5+a6+a7+a8+a9 = 234 --------(1)
Also;
The mean of first five observations is 28.
=> (a1+a2+a3+a4+a5)/5 = 28
=> a1+a2+a3+a4+a5 = 28•5
=> a1+a2+a3+a4+a5 = 140 ------------(2)
Also;
The mean of last five observations is 24.
=> (a5+a6+a7+a8+a9)/5 = 24
=> a5+a6+a7+a8+a9 = 24•5
=> a5+a6+a7+a8+a9 = 120 -----------(3)
Now ,
Adding eq-(1) and (2) , we get;
=> a1+a2+a3+a4+a5+a5+a6+a7+a8+a9 = 140 + 120
=> a1+a2+a3+a4+2•a5+a6+a7+a8+a9 = 260 -----(4)
Now,
Subtracting eq-(1) from (4) , we get;
=> (a1+a2+a3+a4+2•a5+a6+a7+a8+a9)
– (a1+a2+a3+a4+a5+a6+a7+a8+a9) = 260 - 234
=> a1+a2+a3+a4+2•a5+a6+a7+a8+a9
-a1-a2-a3-a4-a5-a6-a7-a8-a9 = 260 - 234
=> a1-a1 + a2-a2 + a3-a3 + a4-a4 + 2•a5-a5
+ a6-a6 + a7-a7 + a8-a8 + a9-a9 = 26
=> a5 = 26
Hence,
The fifth observation is 26 .
Answer:
Mean of 9 observations = 26
Mean of first 5 observation = 28
Mean of last 5 observations = 24
As there are total 9 observations
9 is odd So mid observation is 5 th observation
So, Sum of first 5 and last 5 observations would be sum of all observations + 5 th observation
That is, 5 th observation would appear 2 times as 5 th observation is Mid observation
So,Mean of first 5 observations = 28
Sum of first 5 observation = 28× 5 = 140
Mean of last 5 observation = 24
Sum of last 5 observation = 24×5 = 120
Mean of 9 observation = 26
Sum of 9 observation = 26× 9 = 234
Sum of first 5 + sum of last 5 observation - 5th observation = sum of 9 observation
140+120 - 5th observation = 234
260 - 234 = 5th observation
26 = 5th observation
So, fifth observation is 26
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