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 tim

[tex]\large\underline{\sf{Solution-}}[/tex]

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,

[tex]\rm :\longmapsto\: \: Ayush \: 1 \: day \: work \: = \dfrac{1}{x} [/tex]

[tex]\rm :\longmapsto\: \: Rahul \: 1 \: day \: work = \dfrac{1}{y} [/tex]

[tex]\rm :\longmapsto\: \: Veer \: 1 \: day \: work = \dfrac{1}{z} [/tex]

Now,

According to statement,

• Ayush and Rahul together complete a work in half of time of veer.

So, there 1 day work is

[tex]\rm :\longmapsto\:\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{2}{z} – – – (1)[/tex]

Also, given that

• Ayush and veer together can complete same work in 1/3rd time of Rahul.

So, there 1 day work is

[tex]\rm :\longmapsto\:\dfrac{1}{x} + \dfrac{1}{z} = \dfrac{3}{y} – – – (2)[/tex]

Also, given that

• They together complete a piece of work in 30days.

So, there 1 day work is

[tex]\rm :\longmapsto\:\dfrac{1}{x} + \dfrac{1}{y} + \dfrac{1}{z} = \dfrac{1}{30} [/tex]

[tex]\rm :\longmapsto\:\dfrac{2}{z} + \dfrac{1}{z} = \dfrac{1}{30} \: \: \: \: \: \: \{using \: (1) \}[/tex]

[tex]\rm :\longmapsto\:\dfrac{3}{z} = \dfrac{1}{30} [/tex]

[tex]\bf\implies \:z \: = \: 90 – – – (4)[/tex]

Now,

On Subtracting equation (2) from equation (1), we get

[tex]\rm :\longmapsto\:\dfrac{1}{y} – \dfrac{1}{z} = \dfrac{2}{z} – \dfrac{3}{y} [/tex]

[tex]\rm :\longmapsto\:\dfrac{4}{y} = \dfrac{3}{z} [/tex]

[tex]\rm :\longmapsto\:\dfrac{4}{y} = \dfrac{3}{90} \: \: \: \: \: \: \: \{ \because \: z = 90 \}[/tex]

[tex]\rm :\longmapsto\:\dfrac{4}{y} = \dfrac{1}{30} [/tex]

[tex]\bf\implies \:y = 120[/tex]

[tex]\bf\implies \:Rahul \: alone \: take \: 120 \: days \: to \: finish \: the \: work.[/tex]