Answer:
Juanita Alone will take 9 hours to complete the job.
Solution:
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 = [tex]\frac{W}{6}[/tex]
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 = [tex]\frac{W}{x}[/tex]
Work done by Rick in 1 hour = [tex]\frac{W}{(x + 9)}[/tex]
So when they work together, work done in 1 hour = [tex]\frac{W}{x} + \frac{W}{(x + 9)}[/tex]
Also initially we evaluated that work done by them in 1 hr = [tex]\frac{W}{6}[/tex]
So [tex]\frac{W}{x} + \frac{W}{(x + 9)}[/tex] = [tex]\frac{W}{6}[/tex]
[tex]\mathrm{W}\left(\frac{1}{x}+\frac{1}{x+9}\right)=\frac{W}{6}[/tex] = [tex]\frac{W}{6}[/tex]
[tex]\frac{1}{x}+\frac{1}{x+9}=\frac{1}{6}[/tex]
On cross-multiplication we get
[tex]\frac{(x+9)+x}{x(x+9)}=\frac{1}{6}[/tex]
[tex]\frac{2 x+9}{x^{2}+9 x}=\frac{1}{6}[/tex]
Again on cross-multiplication we get,
[tex]\begin{array}{c}{12 x+54=x^{2}+9 x} \\ {x^{2}-3 x-54=0}\end{array}[/tex]
On splitting the middle term we get
[tex]=x^{2}-9 x+6 x-54=0[/tex]
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.
