Question

It takes two hours for two machines to manufacture 10,000 parts. If Machine #1 can do the job alone in one hour less than Machine #2 can do the job, how long does it take for each machine to manufacture 10,000 parts alone?

Short answer

The time take by machine 2 is $4.6h$and by machine 1 is$3.6h$

Step-by-step solution

Step 1. Given Information 

The given data is it takes 2 hours for two machines to manufacture. Machine 1 do the job alone in 1 hour less than machine 2.

Step 2. Calculation

Let the time taken by machine 2 is $thours$,

Use the given information to complete the table.

Type	No. of hours needed to complete the job	Part of job completed per hour.
Machine 1	$t-1$	$\frac{1}{t-1}$
Machine 2	$t$	$\frac{1}{t}$
Together	2	$\frac{1}{2}$

Since, the part completed by machine 1 plus the part completed by machine 2 equals to the amount completed together. So, we get,

$\frac{1}{t}+\frac{1}{t-1}=\frac{1}{2}$

Solve the equation to find the value of time,

$$\begin{gathered} 2t(t-1)(\frac{1}{t}+\frac{1}{t-1})=\frac{1}{2}\left(2t\right)(t-1) \\ 2t-2+2t=t^2-t \\ {t}^2-t-4t+2=0 \\ {t}^2-5t+2=0 \end{gathered}$$

Use the quadratic formula to simplify further.

$$\begin{gathered} t=\frac{-5\pm \sqrt{{(-5)}^2-4\left(1\right)\left(2\right)}}{2\left(1\right)} \\ t=\frac{5\pm \sqrt{17}}{2} \\ t\approx 0.4,4.6 \end{gathered}$$

Because the $0.4h$does not make any sense so we will discard this value and consider the another one.

Thus, time taken by machine 2 is $4.6h$and machine 1 is$4.6-1=3.6h$