Solve it!

Let $A$ be $x+4$ and $B$ be $x+16$ where $x$ is the number of days they can complete the job together.

If $A$ can complete the job in $x+4$ days, then in $1$ day he would have completed $\frac { 1 }{ x+4 }$ of the job.

And this goes the same for $B$ : $\frac { 1 }{ x+16 }$

Together, they can finish the job in $x$ days, so in $1$ day, they complete $\frac {1}{x}$ of the job.

Equating gives us $\frac { 1 }{ x+4 } +\frac { 1 }{ x+16 } =\frac { 1 }{ x }$ and $\boxed{x=8}$

Let A work at a rate A, and B work at a rate B, and let T be the time for them to complete the job if they both work together. Since in all cases the same amount of work needs to be done we find

$(A+B)T=A(T+4)=B(T+16)$

Divide through by A and let R be the ratio of A's rate to B's rate to get

$(R+1)T=R(T+4)=T+16$

The first and last of these equations give

$R=\frac{16}{T}$

Use this with the second and third equations gives

$\frac{16}{T} \times 4=T$

so (easy!)

$T=8$

We want to know how long it takes to finish a project/job if A and B worked together. Lets call this unknown x days. We are told that A takes 4 more days ( x+4) if work alone, and we are also told that B takes 16 more days( x+16) if work alone. Now lets equate all of these in terms of man/days.

A+B in 1 day can do 1/x th job A in 1 day can do 1/(4+x) th job B in 1 day can do 1/(16+x) th job Therefore, 1/x= 1/(4+x)+1/(16+x) and x= 8

If A+B inn 1 day can do 1/8 th job, they will take 8/8 * 8 days to complete the job ( 8/8 th) or 8 days!

No of days required to finish a job= product of A&B÷ Sum of A&B

Only 8 days satisfies the formula A=4+8=12days & B= 16+8=24 days

X=12×24÷12+24 X= 8 days