| Required run rate =$ \left(\dfrac{282 - (3.2 \times 10)}{40} \right) $=$ \dfrac{250}{40} $ = 6.25 |
Average of 20 numbers = 0.
$\therefore$ Sum of 20 numbers $\left(0 \times 20\right)$ = 0.
It is quite possible that 19 of these numbers may be positive and if their sum is $ a $ then 20th number is $\left(- a \right)$.
Number of members in the team = 11
Let the average age of of the team = $x$
=> $\dfrac{\text{Sum of ages of all 11 members}}{11} = x$
=> Sum of the ages of all 11 members = $11x$
Age of the captain = 26
Age of the wicket keeper = 26 + 3 = 29
Sum of the ages of 9 members of the team excluding captain and wicket keeper
$= 11$x$ - 26 - 29 = 11$x$ - 55$
Average age of 9 members of the team excluding captain and wicket keeper
$=\dfrac{11x - 55 }{9}$
Given that $\dfrac{11x - 55 }{9} = \left(x - 1\right)$
$\Rightarrow 11x - 55 = 9 \left(x - 1\right)$
$\Rightarrow 11x - 55 = 9x - 9$
$\Rightarrow 2x = 46$
$\Rightarrow x = \dfrac{46}{2} = 23\text{ years}$
Total runs scored in 10 matches = $10 \times 38.9$
Total runs scored in first 6 matches =$ 6 \times 42$
Total runs scored in the last 4 matches =$ 10 \times 38.9 - 6 \times 42$
Average of the runs scored in the last 4 matches = $\dfrac{10 \times 38.9 - 6 \times 42}{4}$
$=\dfrac{389 - 252}{4}$
$=\dfrac{137}{4}$
$= 34.25$
| Required average | =$ \left(\dfrac{67 \times 2 + 35 \times 2 + 6 \times 3}{2 + 2 + 3} \right) $ |
| =$ \left(\dfrac{134 + 70 + 18}{7} \right) $ | |
| =$ \dfrac{222}{7} $ | |
| = 31 $ \dfrac{5}{7} $years. |
| Required average | =$ \left(\dfrac{55 \times 50 + 60 \times 55 + 45 \times 60}{55 + 60 + 45} \right) $ |
| =$ \left(\dfrac{2750 + 3300 + 2700}{160} \right) $ | |
| =$ \dfrac{8750}{160} $ | |
| = 54.68 |
Let the average age of the whole team by $ x $ years.
$\therefore 11 x - (26 + 29) = 9\left( x -1 \right)$
$\Rightarrow$ 11$ x $ - 9$ x $ = 46
$\Rightarrow$ 2$ x $ = 46
$\Rightarrow x $ = 23.
So, average age of the team is 23 years.
$\Rightarrow \dfrac{137 + x}{12} = 12\\~\\$
$\Rightarrow 137 + x = 144\\~\\$
$\Rightarrow x = 144 - 137 = 7$
Total runs scored by the player in 40 innings = $ 40 \times 50$
Total runs scored by the player in 38 innings after excluding two innings = $ 38 \times 48$
Sum of the scores of the excluded innings = $ 40 \times 50 - 38 \times 48 = 2000 - 1824 = 176$
Given that the scores of the excluded innings differ by 172. Hence lets take
the highest score as x + 172 and lowest score as $x$
Now $x + 172 + x$ = 176
=> 2$x$ = 4
=> $x=\dfrac{4}{2}$ = 2
Highest score = $x$ + 172 = 2 + 172 = 174
Let Aruns weight by X kg.
According to Arun, 65 < X < 72
According to Aruns brother, 60 < X < 70.
According to Aruns mother, X <= 68
The values satisfying all the above conditions are 66, 67 and 68.
| $\therefore$ Required average =$ \left(\dfrac{66 + 67 + 68}{3} \right) $=$ \left(\dfrac{201}{3} \right) $= 67 kg. |