The average of eleven numbers is 68. The average of the first four numbers is 78 and that of the next four numbers is 63. The $$9^{th}$$ number is two times the $$11^{th}$$ number and the $$10^{th}$$ number is less than the $$11^{th}$$ number. What is the average of the $$9^{th}$$ and $$11^{th}$$ numbers?

Average of $$11^ {th}$$  number  = 68 

sum of $$11 ^ {th}$$ number = $$ 11 \times 68 = 748 $$

average of 1st four number = 78

sum of 1st four number = $$ 78\times 4 $$ = 312 

average of next four number = 63

sum of next four number  = $$63\times 4 $$ = 252 

Let the $$11^{th} $$ number = $$x$$

so $$ 9^{th} $$  number = $$ 2x$$ 

$$10^{th} $$ number is less than $$11^{th}$$ number

$$\Rightarrow  748- (312+ 252) = 184 $$

then $$9^{th}  + $$ 10^{th}+ $$11^{th}= 184$$

$$\Rightarrow 2x + (x-4) + x = 184 $$

$$\Rightarrow 4x - 4 = 184 $$

$$\Rightarrow x = 47 $$

hence Average $$9^{th}$$ and $$11^{th}$$ = $$\dfrac {47 + 2\times 47}{2}$$

$$\Rightarrow \dfrac{47+94}{2}$$

$$\Rightarrow \dfrac{188}{2}$$

$$\Rightarrow 70.5 $$Ans