Hi Kayla,
Let's say for another job each person needs one hour to finish it alone. How long will it take for two people to finish it together? They would finish 1+1 = 2 jobs in one hour or one job in the reciprocal of 1/2 hour.
Now for the real problem . . .
Let X = the time it takes one person to do the job and let X-1 equal the time it takes the other.
So person #1 finishes 1/X of the job in one hour and person #2 finishes 1/(X-1) of the job in one hour.
Together they finish 1/X + 1/(X-1) of the job in one hour.
Multiply the first fraction by (X-1)/(X-1) and multiply the second fraction by X/X to get a common denominator so the sum is: (X-1+X)/)X(X-1)
The time to finish the job together equals the reciprocal of the fraction
(X squared minus X)/(2X-1) = 4
Cross multiply
8X-4 = X squared minus X
Add X to both sides
9X-4 = X squared
Subtract X squared from both sides
-X squared +9X-4 = 0
Multiply everything by -1
X squared - 9X + 4 = 0
Plug into the quadratic equation where a = 1 b = -9 c = 4
X = 9 +/- sq root (81-16)
----------------------------
2
X = (9+8.1)/2 and X = (9-8.1)/2
X = 8.6 or X = 0.5
0.5 can't be correct or else person #2 finishes the job alone in negative time.
So person #1 needs 8.6 hours to finish the job alone and the faster person #2 needs 7.6 hours to finish the job.
