Alex takes 5 days less than Peter to do a project if they work on the project alone. If they are working together, the project would be completed 4 days earlier than Alex working alone. How long would each take to work on the job alone?
This is similar to the last one
Let the number of days it takes Alex to complete the job = D
The Peter takes D + 5 days
So Alex does 1/D of the project in one day
And Peter does 1 / (D + 5) of the project in one day
Add these
1/D + 1/{D + 5) = [ D + 5 + D ] / [ D ( D + 5) ] = [ 2D + 5 ] / [ D^2 + 5D ]
The reciprocal of this is the total time for both to completet the job (in days )= [ D^2 + 5D ] / [ 2D + 5 ]
And like the last problem we know that
Time working together + 4 days = Time for alex working alone
[D^2 + 5D ] / [ 2D + 5 ] + 4 = D simplify
[ D^2 + 5D ] / [ 2D + 5 ] = D - 4
D^2 + 5D = [ 2D + 5] [ D - 4 ]
D^2 + 5D = 2D^2 + 5D - 8D - 20
D^2 - 8D - 20 = 0 factor
(D - 10 ) ( D + 2)
The first factor set to 0 results in D = 10 days for Alex to complete the project on his own
And D + 5 = 15 days for Peter to complete the project on his own
Here is another way:
Peter =1 / P - his work rate
Alex = 1 / [P - 5] - his work rate
1 Job = 1 / J
1 / [P - 5] = 1 / J.....................(1)
1 / P + 1/ [P - 5] = 1 / [J - 4].....(2), solve for P, J
P = 15 days for Peter to finish working by himself.
15 - 5 =10 days for Alex to finish the job by himself.
Working together: [1/15 + 1/10] =1/6 - or 6 days to finish the job. or, 10 - 6 =4 days less than Alex working alone.