Question

The mean of 10 number is 24. If one more number is included, the new mean is 25. Find the included number

Answer 1

I hope it will help u

Answer 2

$<u><b>$
$\bold{Let \: the \: 10 \: numbers \: are \: x_{1} , x_{2} , x_{3} , x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}, x_{10} }$
$</u>$



We know,

$\boxed{ \bold{Mean = \dfrac{sum \: of \: observations}{numbe r \: of \: observations}}}$



Here,in the question ,

$\text{Sum of observations} = \: x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + x_{6} + x_{7} + x_{8} + x_{9} + x_{10} \\ \text{Number of observations = 10 } \\ \text{Mean = 24}$




Hence, by applying the formula of mean we get


$\text{24 }= \dfrac{\: x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + x_{6} + x_{7} + x_{8} + x_{9} + x_{10} }{ 10}$




$24 \times 10 = \: x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + x_{6} + x_{7} + x_{8} + x_{9} + x_{10} \\ \\ 240\: = x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + x_{6} + x_{7} + x_{8} + x_{9} + x_{10} \: \: \: \: \: - - - - - - - : \: ( \: 1 \: )$





$\text{Given that one more is added. }\\ \text{Let the number which id added be} x_{11}$




So,

$\text{Sum of observations} = x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + x_{6} + x_{7} + x_{8} + x_{9} + x_{10} + x_{11} \\ \text{ Number of observations = 10 + 1 = 11} \ \\ \text{ New mean = 25}$


$\text{New mean} = \dfrac{ x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + x_{6} + x_{7} + x_{8} + x_{9} + x_{10} + x_{11}}{ 11 }$




$\text{Putting the value of \:} \bold{( x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + x_{6} + x_{7} + x_{8} + x_{9} + x_{10} ) \: }\text{from ( 1 ) }$



$25 = \dfrac{ 240 + x_{11}}{ 11 } \\ \\ 25 \times 11 = 240 + x _{11} \\ \\ 275 - 240 = x_{11} \\ \\ 35 = x_{11}$





Thus, the number which was included is 35.