The average of twelve numbers is 58. The average of the first five numbers is 56 and the average of the next four numbers is 60. The $$10^{th}$$ number is 4 more than the $$11^{th}$$ number and the $$11^{th}$$ number is one less than the $$12^{th}$$ number. What is the average of the $$10^{th}$$ and $$12^{th}$$ numbers?

Average = $$\frac{\text{sum of the term}}{\text{number of the terms}}$$

Sum of the twelve numbers = 58 $$\times$$ 12 = 696

sum of the first five numbers = 56 $$\times$$ 5 = 280

sum of the next four numbers = 60 $$\times$$  4 = 240

sum of the first nine numbers = 280 + 240 = 520

$$10^{th}$$ number = 4 + $$11^{th}$$ number

$$12^{th}$$ number  = $$11^{th}$$ number + 1

Sum of $$10^{th}$$, 1$$11^{th}$$ and $$12^{th}$$ number = 696 - 520 = 176

$$11^{th}$$ number + 4 + $$11^{th}$$ number + $$11^{th}$$ number + 1 = 176

3 $$\times $$ $$11^{th}$$ number = 176 - 5 = 171

$$11^{th}$$ number = 171/3 = 57

$$10^{th}$$ number = 4 + 57 = 61

$$12^{th}$$ number = 57 + 1 = 58

Average of the $$10^{th}$$ and $$12^{th}$$ numbers = $$\frac{58 + 61}{2}$$ = 59.5