The average of 24 numbers is 56. The average of the first 10 numbers is 71.7 and that of the next 11 numbers is 42. The next three numbers (i.e., $$22^{nd}, 23^{rd}$$ and $$24^{th}$$) are in the ratio $$\frac{1}{2} : \frac{1}{3} : \frac{5}{12}$$. What is the average of the $$22^{nd}  and  24^{th}$$ numbers?

The average of 24 numbers = 56

Sum of the numbers = 56 $$\times$$ 24 = 1344

The average of the first 10 numbers = 71.7

Sum of the first 10 numbers = 71.7 $$\times$$ 10 = 717

The average of the next 11 numbers = 42

Sum of the next 11 numbers = 42 $$\times$$ 11 = 462

Ratio of the next three numbers (i.e., $$22^{nd}, 23^{rd}$$ and $$24^{th}$$) = $$\frac{1}{2} : \frac{1}{3} : \frac{5}{12}$$

Sum of next three numbers (i.e., $$22^{nd}, 23^{rd}$$ and $$24^{th}$$) = 1344 - 717 - 462 = 165

$$22^{nd}$$ number = 165 $$\times \frac{\frac{1}{2}}{\frac{1}{2} + \frac{1}{3} + \frac{5}{12}}$$

=165 $$\times\frac{1/2}{15/12}$$ = 165 $$\times\frac{6}{15}$$ = 66

$$24^{nd}$$ number = 165 $$\times \frac{\frac{5}{12}}{\frac{1}{2} + \frac{1}{3} + \frac{5}{12}}$$

=165 $$\times\frac{5/12}{15/12}$$ = 165 $$\times\frac{1}{3}$$ = 55

Average of the $$22^{nd} and 24^{th}$$ numbers = $$\frac{66 + 55}{2} $$ = 60.5