If $$x = \frac{1}{12.13} + \frac{1}{13.14} +\frac{1}{14.15}+.....+\frac{1}{23.24}   y = \frac{1}{36.37} + \frac{1}{37.38} + \frac{1}{38.39} + .......... + \frac{1}{71.72}$$, then $$\frac{x}{y}$$ is equal to:

$$x = \frac{1}{12.13} + \frac{1}{13.14} +\frac{1}{14.15}+.....+\frac{1}{23.24}$$

$$x=\frac{13-12}{12.13}+\frac{14-13}{13.14}+\frac{15-14}{14.15}+....\frac{24-23}{23.24}$$

$$x=\frac{1}{12}-\frac{1}{13}+\frac{1}{13}-\frac{1}{14}+\frac{1}{14}+\frac{1}{15}+....\frac{1}{23}-\frac{1}{24}$$

$$x=\frac{1}{12}-\frac{1}{24}=\frac{2-1}{24}$$

$$x=\frac{1}{24}$$

$$y = \frac{1}{36.37} + \frac{1}{37.38} + \frac{1}{38.39} + .......... + \frac{1}{71.72}$$

$$y=\frac{37-36}{36.37}+\frac{38-37}{37.38}+\frac{39-38}{38.39}+....\frac{72-71}{71.72}$$

$$y=\frac{1}{36}-\frac{1}{37}+\frac{1}{37}-\frac{1}{38}+\frac{1}{38}+\frac{1}{39}+....\frac{1}{71}-\frac{1}{72}$$

$$y=\frac{1}{36}-\frac{1}{72}=\frac{2-1}{72}$$

$$y=\frac{1}{72}$$

So, $$\frac{x}{y}=\frac{72}{24}=3$$