Vahe and Davit together can complete a task in 3 days. If they had to work separately, the time taken by Davit to complete it would be more than that of Vahe by 8 days. In how many days can Vahe alone complete the task?

When combining rates you are basically adding fractions. so if we say Vahe can do v (as a variable) tasks per 3 days and Davit can complete d tasks per 3 days it would look like [tex]\frac{v}{3} + \frac{d}{3} = \frac{1}{3}[/tex]

now, we know that vahe can complete a task in let's say x days and Davit can complete the task in x+8 days so that means [tex]\frac{1}{x}[/tex] and [tex]\frac{1}{x+8}[/tex]. If we can find x we can make these into the v and d terms above, so we need to find x. If we add these two fractions, like we do the stuff above, we get [tex]\frac{x+8+x}{x(x+8)}[/tex] = [tex]\frac{2x+8}{x(x+8)}[/tex]. so now we can set [tex]\frac{2x+8}{x(x+8)}[/tex] = [tex]\frac{1}{3}[/tex]

from here it's algebra.

[tex]\frac{2x+8}{x(x+8)}[/tex] = [tex]\frac{1}{3}[/tex] Cross multiply

3(2x+8) = x(x+8) expand

6x + 24 = x^2 +8x Get everything to one side

0 = x^2 + 2x -24 Factor however you like, I can go through it if you also want.

You wind up with x = 4

So, now we can plug into the equation with x. So vahe can complete a task in 4 days and Davit 12 days. We can check by adding the fractions [tex]\frac{1}{4} + \frac{1}{12} which gives us that 1/3