Question

Two machines can finish a job in StartFraction 20 Over 9 EndFraction hours. Working​ alone, one machine would take one hour longer than the other to complete the same job. How long would it take the slower machine working alone to complete this​ job?

Answer 1

Answer

5 hours

Explanation

The two working together can finish a job in

$\frac{20}{9} \: hours$

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

$\frac{1}{x} + \frac{1}{x - 1}$

This should be equal to 20/9 hours.

$\frac{1}{x} + \frac{1}{x - 1} = \frac{9}{20}$

Multiply through by;

$20x(x - 1) \times \frac{1}{x} +20x(x - 1) \times \frac{1}{x - 1} = 20x(x - 1) \times \frac{9}{0}$

$20(x - 1) +20x = 9x(x - 1)$

$20x - 20+20x = 9{x}^{2} - 9x$

$9{x}^{2} - 9x - 20x - 20x + 20= 0$

$9{x}^{2} - 49x + 20= 0$

Factor to get:

$(9x - 4)(x - 5) = 0$

$x = \frac{4}{9} \: or \: x = 5$

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.