[tex]\large\underline{\sf{Solution-}}[/tex]

Given that, the mean of 6 numbers is 24.

Number of observations, n = 6

Mean of 6 observations = 24

We know,

[tex]\boxed{ \sf{ \:\sf \:Mean = \dfrac{Sum \: of \: observations}{Number \: of \: observations } \: }} \\ \\ [/tex]

So,

Sum of 6 observations = Number of observations × Mean

Sum of 6 observations = 6 × 24 = 144

Let assume that excluded number be x.

So, Number of observations = 5

Sum of 5 observations = 144 - x

Mean of 5 observations = 22

We know,

[tex]\boxed{ \sf{ \:\sf \:Mean = \dfrac{Sum \: of \: observations}{Number \: of \: observations } \: }} \\ \\ [/tex]

So, on substituting the values, we get

[tex]\sf \: 22 = \dfrac{144 - x}{5} \\ \\ [/tex]

[tex]\sf \: 110 = 144 - x \\ \\ [/tex]

[tex]\sf \: x = 144 - 110 \\ \\ [/tex]

[tex]\sf \:  \implies \: x = 34 \\ \\ [/tex]

So, excluded number is 34

Answer:

Given :-

• The mean of 6 numbers is 24.
• One number is excluded, the mean of remaining number becomes 22.

To Find :-

• What is the excluded number.

Solution :-

Let,

[tex]\mapsto \bf Excluded\: Number =\: x\\[/tex]

First, we have to find the sum of 6 observations :

[tex]\bigstar[/tex] The mean of 6 numbers is 24.

Given :

• Mean = 24
• Number of Observations = 6

According to the question by using the formula we get,

[tex]\small \implies \sf\boxed{\bold{Mean =\: \dfrac{Sum\: Of\: Observations}{Total\: Number\: Of\: Observations}}}\\[/tex]

So, by putting those values we get,

[tex]\implies \sf 24 =\: \dfrac{Sum\: Of\: Observations}{6}\\[/tex]

By doing cross multiplication we get,

[tex]\implies \sf Sum\: Of\: Observations =\: 24(6)\\[/tex]

[tex]\implies \sf Sum\: Of\: Observations =\: 24 \times 6\\[/tex]

[tex]\implies \sf\bold{Sum\: Of\: Observations =\: 144}\\[/tex]

Hence, the sum of 6 observations is 144 .

Now,

[tex]\bigstar[/tex] If one number is excluded, the mean of remaining number becomes 22.

So,

[tex]\leadsto \sf New\: Sum\: Of\: Observations =\: 144 - x\\[/tex]

[tex]\leadsto \sf Total\: Number\: Of\: Observations =\: 6 - 1 =\: 5\\[/tex]

Given :

• New Sum Of Observations = 144 - x
• Total Number Of Observations = 5
• New Mean = 22

According to the question by using the formula we get,

[tex]\small \implies \sf\boxed{\bold{New\: Mean =\: \dfrac{New\: Sum\: Of\: Observations}{Total\: Number\: Of\: Observations}}}\\[/tex]

By putting those values we get,

[tex]\implies \sf 22 =\: \dfrac{144 - x}{5}\\[/tex]

By doing cross multiplication we get,

[tex]\implies \sf 144 - x =\: 22(5)[/tex]

[tex]\implies \sf 144 - x =\: 22 \times 5[/tex]

[tex]\implies \sf 144 - x =\: 110[/tex]

[tex]\implies \sf - x =\: 110 - 144[/tex]

[tex]\implies \sf {\cancel{-}} x =\: {\cancel{-}} 34[/tex]

[tex]\implies \sf\bold{x =\: 34}[/tex]

Hence, the required excluded number will be :

[tex]\dag[/tex] Excluded Number :

[tex]\dashrightarrow \sf Excluded\: Number =\: x\\[/tex]

[tex]\dashrightarrow \sf\bold{\underline{Excluded\: Number =\: 34}}\\[/tex]

[tex]\therefore[/tex] The excluded number is 34 .