Question

the mean of 7 numbers is 24 if out of these 7 numbers the mean of first four number is 23 and the mean of the last four number is 25.5 then find the fourth number

Answer 1

$\text{Let the 7 numbers be}x_1,\;x_2,\;x_3,\;x_4,\;x_5,\;x_6,\;x_7$

$\textbf{Given:}$

$\text{Mean of the 7 numbers = 24}$

$\displaystyle\frac{x_1+x_2+x_3+x_4+x_5+x_6+x_7}{7}=24$

$\displaystyle\,x_1+x_2+x_3+x_4+x_5+x_6+x_7=168$........(1)

$\bf\text{Mean of the first 4 numbers = 23}$

$\displaystyle\frac{x_1+x_2+x_3+x_4}{4}=23$

$\displaystyle\,x_1+x_2+x_3+x_4=92$........(2)

$\bf\text{Mean of the last 4 numbers = 25.5}$

$\displaystyle\frac{x_4+x_5+x_6+x_7}{4}=25.5$

$\displaystyle\,x_4+x_5+x_6+x_7=102$........(3)

$\text{Adding (2) and (3), we get}$

$\displaystyle\,x_1+x_2+x_3+2(x_4)+x_5+x_6+x_7=194$

$\displaystyle\,(x_1+x_2+x_3+x_4+x_5+x_6+x_7)+x_4=194$

$\implies\displaystyle\,(168)+x_4=194$

$\implies\displaystyle\,x_4=194-168$

$\implies\displaystyle\,x_4=26$

$\therefore\textbf{The fourth number is 26}$