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If Person A can deliver papers in 40 min, and person B can do the same papers in 50 min, how long does it take when they work together?

This is a rational expression problem. Is there an easy, or not complicated way to do this?

$\dfrac1{40} + \dfrac1{50} = \dfrac1x $

It didn't work this way though. It did for a different problem. Why must I keep changing the format? Why can't this format work?

hola
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Jake
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    What are your thoughts about this? The solution is indeed not complicated. But where do you get stuck? – mickep Jan 15 '15 at 08:42
  • I tried doing it this way: 1 Paper Route/ 40 mins + 1 Paper Route/ 50 mins equals to 1 Paper Route/ x minutes. – Jake Jan 15 '15 at 09:10
  • @Jake What do you mean by 'work together'? Two person can work together just by gossiping (and not doing anything worthy). – hola Jan 15 '15 at 09:32

2 Answers2

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Consider $P$ the amount of papers. In one minute the first person delivers $\frac{P}{40}$, the second one $\frac{P}{50}$. Together in one minute they deliver $\frac{P}{40}+\frac{P}{50}$. So they need

$$\frac{P}{ \frac{P}{40}+\frac{P}{50}}$$ minutes to deliver the whole thing.

The quantity $P$ magically disappears...

orangeskid
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  • I know this may sound a little ignorant, but, why can't they just spend 45 minutes on it together, since one takes 40 minutes, and the other, 50? Is it because of the different rate of work each one does? – Jake Jan 15 '15 at 09:16
  • Jake: yes, the faster worker takes on more of the work. If one took 10 min and the other 60, obviously it wouldn't take 35 mins working together. – HTFB Jan 15 '15 at 09:21
  • @Jake: The faster, helped by the slower, gets done even quicker. The time is $\frac{200}{9}$, almost half of $40$. Also: if both were equally fast the time would halve. In general, it's even better than $\it half$ the average in fact. – orangeskid Jan 15 '15 at 09:21
  • @Jake: Somehow the idea is to see how much they make per minute. If you are more comfortable with providing a value for $P$ do so, it may help at the beginning. It's all good. – orangeskid Jan 15 '15 at 09:24
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In $200$ minutes working together they deliver $5+4=9$ papers. So how many minutes they need for delivering $1$ paper?

drhab
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