SOLUTION: One machine takes 15 minutes to do a certain job. Another machine takes 20 minutes to do the same job. Find the time it would take both machines, working together, to do the same

Question 106205: One machine takes 15 minutes to do a certain job. Another machine takes 20 minutes to do the same job. Find the time it would take both machines, working together, to do the same job. Be sure to define your variable, write an equation, and show how you solved the equation. Please help.
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
Since the first machine does a job in 15 minutes, then each minute it does one-fifteenth
of the job. Meanwhile, the second machine does the same job in 20 minutes.
Therefore, each minute that goes by it does one-twentieth of the job. This means
that when the machines are working together, each minute that goes by the combined work that
they do is of the job. You can add these two numbers together by converting
them to a common denominator of 60 and then adding them. and .
Therefore, adding them together results in .
.
So each minute that goes by the two machines do seven-sixtieths of the one job.
.
We can solve for the number of minutes it takes by letting t represent the variable time.
If you multiply t times and set that equal to 1 job the equation becomes:
.

.
You can solve for t by dividing both sides of this equation by . When you do, the
left side becomes just t, and the right side is .
.
So on the right side you are dividing 1 by . But remember that when you divide
by a fraction, you invert the fraction and multiply it by the number you are dividing.
Therefore the right side becomes and this is equal to 60 divided by 7.
If you do this division you get an answer 8.571428571 seconds or you can round this to
8.57 seconds or 8.6 seconds. So when the two machines are working together, they do
the 1 job in 8.57 seconds.
.
If you have a teacher that wants a little less thinking and a little more equation
work you can start with the equation:
.
= 1
.
Then factor out the t on the left side to get:
.

.
Now combine the and the as was done above. You get
and the equation becomes:
.

.
Then solve for t by dividing both sides of this equation by just as we did
above. You should again get 8.57 minutes (8 minutes & 34.2 seconds) as the answer.
.
Hope this helps you to understand the problem.
.
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