Question

The mean is nine numbers is 77. If one more number is added to it then the mean increases
by 5. Find the number added in the data.


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Answer 1

As We know that the mean of nine numbers is 77 so that :-

$\circ \ {\pmb{\underline{\boxed{\sf{ ( \overline{x} ) = \dfrac{x_1,x_2 \cdots x_n}{n} }}}}} \\ \\ \\ \colon\implies{\sf{ 77 = \dfrac{x_1,x_2 \cdots x_9}{9} }} \\ \\ \\ \colon\implies{\sf{ 693 = ( x_1,x_2 \cdots x_9) \ \ \ \ \ \cdots(1) }}$

Now, If one more number is added to it then the mean increases by 5 as:

Let that number be x

$\colon\implies{\sf{ 77+5 = \dfrac{(x_1,x_2 \cdots x_9)+x}{9+1} }} \\ \\ \\ \colon\implies{\sf{ 82 = \dfrac{(x_1,x_2 \cdots x_9)+x}{10} }} \\ \\ \\ \colon\implies{\sf{ 820 = (x_1,x_2 \cdots x_9)+x }} \\ \\ \\ \colon\implies{\sf{ 820 = 693+x \ \ \ \ \ \pink{[From \ Equation \ (1)]} }} \\ \\ \\ \colon\implies{\sf{ 820 - 693 = x }} \\ \\ \\ \colon\implies{\underline{\boxed{\sf{ x = 127 }}}}$

Hence,

$\\ {\pmb{\underline{\sf\red{ The \ number \ that \ was \ added \ in \ the \ data \ is \ 127. }}}}$