<u>Answer:</u>
Juanita Alone will take 9 hours to complete the job.
<u>
Solution:
</u>
For sake of simplicity let’s assume complete job be represented by W.
Job done by Juanita and Rick together in 6 hours is complete job = W
So job done by Juanita and Rick together in 1 hour = ![\frac{W}{6}](https://tex.z-dn.net/?f=%5Cfrac%7BW%7D%7B6%7D)
Lets assume number of hours needed by Juanita to complete W work = x hrs
And since Rick takes 9 hours more than Juanita , so number of hours needed by Juanita to complete W work = (x + 9) hrs
Work done by Juanita in 1 hour = ![\frac{W}{x}](https://tex.z-dn.net/?f=%5Cfrac%7BW%7D%7Bx%7D)
Work done by Rick in 1 hour = ![\frac{W}{(x + 9)}](https://tex.z-dn.net/?f=%5Cfrac%7BW%7D%7B%28x%20%2B%209%29%7D)
So when they work together, work done in 1 hour = ![\frac{W}{x} + \frac{W}{(x + 9)}](https://tex.z-dn.net/?f=%5Cfrac%7BW%7D%7Bx%7D%20%2B%20%5Cfrac%7BW%7D%7B%28x%20%2B%209%29%7D)
Also initially we evaluated that work done by them in 1 hr = ![\frac{W}{6}](https://tex.z-dn.net/?f=%5Cfrac%7BW%7D%7B6%7D)
So
= ![\frac{W}{6}](https://tex.z-dn.net/?f=%5Cfrac%7BW%7D%7B6%7D)
= ![\frac{W}{6}](https://tex.z-dn.net/?f=%5Cfrac%7BW%7D%7B6%7D)
![\frac{1}{x}+\frac{1}{x+9}=\frac{1}{6}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7Bx%7D%2B%5Cfrac%7B1%7D%7Bx%2B9%7D%3D%5Cfrac%7B1%7D%7B6%7D)
On cross-multiplication we get
![\frac{(x+9)+x}{x(x+9)}=\frac{1}{6}](https://tex.z-dn.net/?f=%5Cfrac%7B%28x%2B9%29%2Bx%7D%7Bx%28x%2B9%29%7D%3D%5Cfrac%7B1%7D%7B6%7D)
![\frac{2 x+9}{x^{2}+9 x}=\frac{1}{6}](https://tex.z-dn.net/?f=%5Cfrac%7B2%20x%2B9%7D%7Bx%5E%7B2%7D%2B9%20x%7D%3D%5Cfrac%7B1%7D%7B6%7D)
Again on cross-multiplication we get,
![\begin{array}{c}{12 x+54=x^{2}+9 x} \\ {x^{2}-3 x-54=0}\end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bc%7D%7B12%20x%2B54%3Dx%5E%7B2%7D%2B9%20x%7D%20%5C%5C%20%7Bx%5E%7B2%7D-3%20x-54%3D0%7D%5Cend%7Barray%7D)
On splitting the middle term we get
![=x^{2}-9 x+6 x-54=0](https://tex.z-dn.net/?f=%3Dx%5E%7B2%7D-9%20x%2B6%20x-54%3D0)
x( x – 9) +6( x – 9 ) = 0
(x+6) (x-9) = 0
When x + 6 = 0, x = -6
When x – 9 = 0, x = 9
Since x is number of hours, it cannot be negative in given case. So required solution is x = 9.
Hence Juanita Alone will take 9 hours to complete the job.