The average of twelve numbers is 45.5. The averageof the first four numbers is 41.5 and that of the next five numbers is 48. The $$10^{th}$$ numberis 4 more than the $$11^{th}$$ number and 9 more than the $$12^{th}$$ number. Whatis the average of the $$10^{th}$$ and $$12^{th}$$ numbers?

The average of 12 numbers = 45.5

Sum of 12 numbers = average $$\times numbers = 45.5 \times 12$$ = 546

The average of the first four numbers = 41.5

Sum of first four numbers = 41.5 $$\times$$ 4 = 166

The average of next five numbers = 48

Sum of next five numbers = 48 $$\times$$ 5 = 240

Sum of the first 9 numbers = sum of first four numbers + sum of next five numbers = 166 + 240 = 406

10th number = 4 + 11th number ---(1)

10th number = 9 + 12th number ---(2)

Sum of the first 9 numbers +10th number + 11th number + 12th number = 546

406 +  10th number +  10th number - 4 + 10th number - 9 = 546

3 $$\times$$ 10th number = 546 - 406 + 4 + 9  

3 $$\times$$ 10th number = 153

10th number = 51

From eq(1),

11th number = 10th number - 4 = 51 - 4 = 47

12th number = 10th number - 9 = 51 - 9 = 42

The average of the $$10^{th}$$ and $$12^{th}$$ numbers = $$\frac{51 + 47 + 42}{3} = 140/3 = 46.6