Ayush and Rahul together complete a work in half of time of veer,while

1.	Ayush and Rahul together complete a work in half of time of veer,while Ayush and veer together can complete same work in 1/3rd time of Rahul .if they together complete a piece of work in 30days.then in how many days Rahul can alone complete the work.
Answer» $\large\underline{\sf{Solution-}}$ Let Time taken by AYUSH alone to FINISH the work be 'x' days. Time taken by Rahul alone to finish the work be 'y' days. Time taken by VEER alone to finish the work be 'z' days. So, $\rm :\longmapsto\: \: Ayush \: 1 \: day \: work \: = \dfrac{1}{x}$ $\rm :\longmapsto\: \: Rahul \: 1 \: day \: work = \dfrac{1}{y}$ $\rm :\longmapsto\: \: Veer \: 1 \: day \: work = \dfrac{1}{z}$ Now, According to statement, Ayush and Rahul together COMPLETE a work in half of time of veer. So, there 1 day work is $\rm :\longmapsto\:\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{2}{z} - - - (1)$ Also, given that Ayush and veer together can complete same work in 1/3rd time of Rahul. So, there 1 day work is $\rm :\longmapsto\:\dfrac{1}{x} + \dfrac{1}{z} = \dfrac{3}{y} - - - (2)$ Also, given that They together complete a PIECE of work in 30days. So, there 1 day work is $\rm :\longmapsto\:\dfrac{1}{x} + \dfrac{1}{y} + \dfrac{1}{z} = \dfrac{1}{30}$ $\rm :\longmapsto\:\dfrac{2}{z} + \dfrac{1}{z} = \dfrac{1}{30} \: \: \: \: \: \: \{using \: (1) \}$ $\rm :\longmapsto\:\dfrac{3}{z} = \dfrac{1}{30}$ $\bf\implies \:z \: = \: 90 - - - (4)$ Now, On Subtracting equation (2) from equation (1), we get $\rm :\longmapsto\:\dfrac{1}{y} - \dfrac{1}{z} = \dfrac{2}{z} - \dfrac{3}{y}$ $\rm :\longmapsto\:\dfrac{4}{y} = \dfrac{3}{z}$ $\rm :\longmapsto\:\dfrac{4}{y} = \dfrac{3}{90} \: \: \: \: \: \: \: \{ \because \: z = 90 \}$ $\rm :\longmapsto\:\dfrac{4}{y} = \dfrac{1}{30}$ $\bf\implies \:y = 120$ $\bf\implies \:Rahul \: alone \: take \: 120 \: days \: to \: finish \: the \: work.$