Answer
5 hours
Explanation
The two working together can finish a job in
[tex] \frac{20}{9} \: hours[/tex]
Also, working alone, one machine would take one hour longer than the other to complete the same job.
Let the slower machine working alone take x hours. Then the faster machine takes x-1 hours to complete the same task working alone.
Their combined rate in terms of x is
[tex] \frac{1}{x} + \frac{1}{x - 1} [/tex]
This should be equal to 20/9 hours.
[tex] \frac{1}{x} + \frac{1}{x - 1} = \frac{9}{20} [/tex]
Multiply through by;
[tex] 20x(x - 1) \times \frac{1}{x} +20x(x - 1) \times \frac{1}{x - 1} = 20x(x - 1) \times \frac{9}{0} [/tex]
[tex]20(x - 1) +20x = 9x(x - 1)[/tex]
[tex]20x - 20+20x = 9{x}^{2} - 9x[/tex]
[tex]9{x}^{2} - 9x - 20x - 20x + 20= 0[/tex]
[tex]9{x}^{2} - 49x + 20= 0[/tex]
Factor to get:
[tex](9x - 4)(x - 5) = 0[/tex]
[tex]x = \frac{4}{9} \: or \: x = 5[/tex]
It is not feasible for the slower machine to complete the work alone in 4/9 hours if the two will finish in 20/9 hours.
Therefore the slower finish in 5 hours.
