Question 403408: If Mike takes 3 hours to make one bicycle and Henry takes 4 hours to make one bicycle, how many hours will it take them to make 7 bicycles together? (I CANNOT set this up correctly, and would appreciate any advice you would share. Thanks for your time.)
Found 2 solutions by jim_thompson5910, robertb:
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! First figure out how long it takes them to make ONE bicycle. If you get stuck with problems like this, always remember that you must start with one (if you want to get to 7)
Since "Mike takes 3 hours to make one bicycle", we know that his rate is  bikes an hour. In other words, in one hour, he has one-third of a bike done.
Also, we know that "Henry takes 4 hours to make one bicycle", so by similar logic his rate is  bikes per hour
Now add their two rates. This is of course assuming ideal conditions.
So under perfect conditions (ie there's no overlap between the two workers, they're working consistently, etc) their combined rate is  bikes an hour. Stated another way, in one hour, together they've completed of a single bike.
But we want a WHOLE bike, not of a bike. So we WANT to obtain the value 1. How do we this? We just let some time pass of course. If we let the two workers keep working, they'll get 1 bike done eventually (no matter how slow they go)
So let's let some unknown time pass. We'll call this time 't'. If t = 1, then 1 hour passes and they've completed of a single bike. Now say 2 hours pass, which means t = 2 now. So this means they've completed ANOTHER of that same bike. Now add the fractions to get 
Notice how  is clearly over 1. So the two workers have completed bike #1 and have moved onto bike #2 in two hours. So we know for sure that the first bike took under 2 hours to make.
In general, given 't' hours, the two workers will make  bikes. We just don't know 't' (yet)
Since we're forcing the number of bikes to be 1, this means that we're setting equal to 1 to get
which is the equation we want to solve.
To do that, multiply both sides by 12 and then divide both sides by 7 to get  (note: you can just flip to get  )
Note: 
So it takes or  hours to make ONE bike...phew. With me so far?
If not, then go back over all of the above (and ask me about anything before moving on)
Since we want 7 bikes, we can just scale up (again assuming everything stays the same). So just multiply by 7 to get 
So it will take 12 hours to make 7 bikes. Note: the book most likely chose these numbers to make the answer clean.
Answer by robertb(5830)  (Show Source):
You can put this solution on YOUR website! The equation is 
Notice the right-hand side of the equation is not equal to 1, because we're not dealing with only 1 WHOLE quantity, but 7 WHOLE quantities.
==>  <==> x = 12 hours.
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