Answer:
Dwight will take 6 hours to finish the job alone and Joe will take 3 hours to finish the job alone.
Step-by-step explanation:
Let us assume the time taken by Dwight to split a cord of firewood = K hrs
So, the per hour rate of Dwight = ![(\frac{1}{K})](https://tex.z-dn.net/?f=%28%5Cfrac%7B1%7D%7BK%7D%29)
As, Joe uses 3 LESS hours then Dwight.
So, the time taken by Joe to split a cord of firewood = (K- 3) hrs
So, the per hour rate of Joe = ![(\frac{1}{K-3})](https://tex.z-dn.net/?f=%28%5Cfrac%7B1%7D%7BK-3%7D%29)
Now, when both of them wok together, it takes them 2 hours.
So, the per hour rate of BOTH of them = ![(\frac{1}{2})](https://tex.z-dn.net/?f=%28%5Cfrac%7B1%7D%7B2%7D%29)
⇒ Per hour rate of ( Dwight + Joe) =![(\frac{1}{2} )](https://tex.z-dn.net/?f=%28%5Cfrac%7B1%7D%7B2%7D%20%29)
![\implies (\frac{1}{K}) + (\frac{1}{K-3}) = (\frac{1}{2})](https://tex.z-dn.net/?f=%5Cimplies%20%28%5Cfrac%7B1%7D%7BK%7D%29%20%2B%20%20%28%5Cfrac%7B1%7D%7BK-3%7D%29%20%3D%20%28%5Cfrac%7B1%7D%7B2%7D%29)
Now, solving for the value of K , we get:
![(\frac{1}{K}) + (\frac{1}{K-3}) = (\frac{1}{2})\\\implies \frac{(K-3) + K}{K (K-3)} = (\frac{1}{2})\\\implies 2(2K -3) = K^2 - 3K\\\implies k^2 - 3K -4K +6 = 0\\\implies K^2 - 7K + 6= 0\\\implies K^2 - 6K - K + 6= 0\\\implies K(K-6) -1(K - 6)= 0\\\implies (K-6)(K-1) = 0](https://tex.z-dn.net/?f=%28%5Cfrac%7B1%7D%7BK%7D%29%20%2B%20%20%28%5Cfrac%7B1%7D%7BK-3%7D%29%20%3D%20%28%5Cfrac%7B1%7D%7B2%7D%29%5C%5C%5Cimplies%20%20%5Cfrac%7B%28K-3%29%20%2B%20K%7D%7BK%20%28K-3%29%7D%20%20%3D%20%28%5Cfrac%7B1%7D%7B2%7D%29%5C%5C%5Cimplies%202%282K%20-3%29%20%3D%20K%5E2%20-%203K%5C%5C%5Cimplies%20k%5E2%20-%203K%20-4K%20%2B6%20%3D%200%5C%5C%5Cimplies%20K%5E2%20-%207K%20%20%2B%206%3D%20%200%5C%5C%5Cimplies%20K%5E2%20-%206K%20-%20K%20%20%2B%206%3D%20%200%5C%5C%5Cimplies%20K%28K-6%29%20-1%28K%20-%206%29%3D%20%200%5C%5C%5Cimplies%20%28K-6%29%28K-1%29%20%3D%200)
Implies either K = 6 Or K = 1
But if K = 1, (K-3) = 1- 3 = -2 hours would be A CONTRADICTION.
⇒ K = 6 hours
Hence, Dwight will take 6 hours to finish the job alone and Joe will take (k-3) = (6-3) = 3 hours to finish the job alone.