Question

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

Answer 1

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 2

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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