Bihar Police Constable Model Paper 5

Given:
The average of thirteen number = 9
The average of the first seven numbers = 10
The average of the last seven numbers = 9
Formula used:
The sum of total numbers = the average of all numbers × total numbers
Calculation:
Let the thirteen number be ${\mathit{x}}_1,{\mathit{x}}_2,{\mathit{x}}_3,{\mathit{x}}_4,{\mathit{x}}_5$, ${\mathit{x}}_6,{\mathit{x}}_7,{\mathit{x}}_8,{\mathit{x}}_9,{\mathit{x}}_{10}$, ${\mathit{x}}_{11},{\mathit{x}}_{12}$ and ${\mathit{x}}_{13}$
The sum of the first seven numbers = 10 × 7
⇒ ${\mathit{x}}_1+{\mathit{x}}_2+{\mathit{x}}_3+{\mathit{x}}_4+{\mathit{x}}_5$ + ${\mathit{x}}_6+{\mathit{x}}_7$ = 70 ----(i)
The sum of the last seven numbers = 9 × 7
⇒ ${\mathit{x}}_7+{\mathit{x}}_8+{\mathit{x}}_9+{\mathit{x}}_{10}+{\mathit{x}}_{11}$ + ${\mathit{x}}_{12}+{\mathit{x}}_{13}$ = 63 ----(ii)
The sum of thirteen numbers = 13 × 9
⇒ ${\mathit{x}}_1+{\mathit{x}}_2+{\mathit{x}}_3+{\mathit{x}}_4+{\mathit{x}}_5$ + ${\mathit{x}}_6+{\mathit{x}}_7+{\mathit{x}}_8+{\mathit{x}}_9+{\mathit{x}}_{10}+{\mathit{x}}_{11}$ + ${\mathit{x}}_{12}+{\mathit{x}}_{13}$ = 117 ----(iii)
After adding equation (i) and (ii), we get,
${\mathit{x}}_1+{\mathit{x}}_2+{\mathit{x}}_3+{\mathit{x}}_4+{\mathit{x}}_5$ + ${\mathit{x}}_6+{\mathit{x}}_7+{\mathit{x}}_8+{\mathit{x}}_9+{\mathit{x}}_{10}+{\mathit{x}}_{11}$ + ${\mathit{x}}_{12}+{\mathit{x}}_{13}$ = 70 + 63
⇒ ${\mathit{x}}_1+{\mathit{x}}_2+{\mathit{x}}_3+{\mathit{x}}_4+{\mathit{x}}_5$ + ${\mathit{x}}_6+2{\mathit{x}}_7+{\mathit{x}}_8+{\mathit{x}}_9+{\mathit{x}}_{10}+{\mathit{x}}_{11}$ + ${\mathit{x}}_{12}+{\mathit{x}}_{13}$ = 133 ----(iv)
After subtracting equation (iii) from equation (iv), we get
${\mathit{x}}_7$ = 133 – 117
⇒ 16
∴ The 7th number is 16.