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I'm not sure if these types of questions are accepted here or not (I'm very sorry if it's not), but it would be great if anyone could explain me this.

Question: Using his bike, Daniel can complete a paper route in 20 minutes. Francisco, who walks the route, can complete it in 30 minutes. How long will it take the two boys to complete the route if they work together, one starting at each end of the route?

I have the answer: 12 minutes

But I don't understand the solution given in the book.

Can any of you explain how to solve this? Your help is highly appreciated.

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Daniel finishes $\frac{1}{20}$ th of his work in 1 min. Fransico finishes $\frac{1}{30}$ th of his work in 1 min. In one min (simultaneously), they finish off $\frac{1}{30}$ + $\frac{1}{20}$ = $\frac{1}{12}$ of the work (Assuming no dependency which is true in this case as they are starting from opposite ends). So, in one minute, they finish off $\frac{1}{12}$ th of the work. So, in 12 minutes, they will finish off the entire work.

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Suppose the paper route is 1 mile in length. Then Daniel is traveling at 3 miles an hour and Fransico is traveling at 2 miles an hour. Imagine the paper route is a straight line running left to right. Daniel starts his route at the far left side of the line traveling towards the right, and Fransico starts his route from the far right side of the line traveling left. We want to know how long it takes for them to meet.

Daniel's position on the line, X, is a function of his speed and time.

X_Daniel = speed * time.

Fransico's position is also a function of time, but he's coming from the right. Since the route is 1 mile long, we subtract his position from 1.

X_Fransico = 1 - speed * time.

Since we want to know the point at which they meet, we set the two functions equal to each other and solve for time.

$3t = 1 - 2t$

$3t + 2t = 1$

$5t = 1$

$t = 1 / 5$ of an hour, or if we divide 60 minutes by 5, we get 12 minutes.

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Suppose that $N$ newspapers have to be delivered. Since Daniel can do the job in $20$ minutes, he distributes $\frac{N}{20}$ newspapers per minute.

Similarly, Francisco delivers $\frac{N}{30}$ newspapers per minute.

So if they work together as described, they deliver a total of $\frac{N}{20}+\frac{N}{30}$ newspapers per minute. In other words, their combined delivery rate is $\frac{N}{20}+\frac{N}{30}$ newspapers per minute.

Thus the total time that they take is the total number of newspapers to be delivered, divided by their combined rate. This is $$\frac{N}{\frac{N}{20}+\frac{N}{30}}.\tag{$1$}$$ Now we need to do some algebra. The denominator in the above expression is $\frac{N}{20}+\frac{N}{30}$. Bring this expression to the common denominator $60$. We have $\frac{N}{20}+\frac{N}{30}=\frac{3N}{60}+\frac{2N}{60}=\frac{5N}{60}=\frac{N}{12}$. So the expression $(1)$ simplifies to $$\frac{N}{\frac{N}{12}},$$ which simplifies to $12$.

Remark: The above calculation has an abstract character. To make it very concrete, decide arbitrarily on the number of newspapers to be delivered. It is convenient to assume there are $60$ papers, because $60$ is divisible by both $20$ and $30$.

If there are $60$ papers to be delivered, then Daniel delivers $60/20$ newspapers per minute, and Francisco delivers $60/30$ newspapers per minute. So their combined delivery rate is $5$ papers per minute. Since there are $60$ papers, it takes $60/5=12$ minutes for the two people to deliver them all.

The first calculation that we made uses the general "$N$" instead of the specific (and possibly wrong) $60$. Apart from that, it is exactly the same as our concrete calculation with $N=60$. It is very useful to go through the calculation with concrete numbers, to see what's really going on.

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