Question

Two friends work together on an assignment rahul and ajay. they complete the assignment together in 15 days.but if rahul works alone and completes half assignment himself and then rest of the half assignment is completed by ajay alone, then the assignment is completed in 40 days. how long will rahul take to complete the assignment alone?

Answer 1

Rahul will take either 60 or 20 days to complete the assignment alone.

Lets assume, Rahul will take x days and Ajay will take y days to complete the whole assignment alone.

They complete the assignment together in 15 days.

So, work done by Rahul alone in 1 day = $\frac{1}{x}$

Work done by Ajay alone in 1 day = $\frac{1}{y}$

and work done by together in 1 day = $\frac{1}{15}$

Thus we will get the equation: $\frac{1}{x}+\frac{1}{y} =\frac{1}{15}$

Rahul will complete half assignment in $\frac{x}{2}$ days and Ajay will complete the remaining half assignment in $\frac{y}{2}$ days. Then the assignment is completed in 40 days.

So, the second equation will be: $\frac{x}{2}+\frac{y}{2}= 40$ or $x+y=80$

Now we will solve system of equations-

$\frac{1}{x}+\frac{1}{y} =\frac{1}{15}$ .................. (1)

$x+y=80$ ................................ (2)

First, we will solve the equation (2) for y itself.

$x+y=80\\ \\ y= 80-x$

Now, we have to substitute this $y=80-x$ into the equation (1):

$\frac{1}{x}+\frac{1}{80-x} =\frac{1}{15}\\ \\ \frac{80-x+x}{x(80-x)}=\frac{1}{15}\\ \\ \frac{80}{x(80-x)}=\frac{1}{15}\\ \\ x(80-x)= 1200\\ \\ 80x-x^2 =1200\\ \\ x^2 -80x+1200=0\\ \\ (x-60)(x-20)=0\\ \\ x= 60 , 20$

So, Rahul will take either 60 or 20 days to complete the assignment alone.