A question from my homework: Two MSU cross-country runners are 440 yards apart and are running towards each other, one at 8mph and the other at 10 mph. In how many seconds will they meet? How do I...

First, we note that 440 yards is equivalent to 0.25 miles. 

To answer the problem, we first try to visualize what's happening.

There are two runners, A and B. The speed of A is 8mph, that of B is 10 mph. A and B are standing on opposite ends of a track that is 0.25 miles long. They are running towards each other. Now, since they are running towards each other, they will definitely meet. The point of meeting is important -- it will be at the same point (of course) and they would have ran after the same amount of time (of course, assuming they started at the same time).

A --------x-------------------B

In the illustration above, x would be the meeting point. Obviously, B would have covered more distance since he has a greater speed. Assuming A covered a distance of x miles, B would have covered a distance of 0.25 - x miles (since the total distance is 0.25 miles. Also, note that we could have assigned x to B and 0.25-x to A but that would not affect the answer). Then, we let the total time lapsed equal to y. The time is equal for both runners.

To summarize our variables:

Speed of A: 8 mph

Speed of B: 10 mph

Distance Travelled by A: x miles

Distance Travelled by B: 0.25 - x miles

Time A Ran: y

Time B Ran: y

Next, we note that speed is simply distance over time. From the given, we can obtain two equations:

`x/y = 8`

`(0.25 - x )/ y = 10`

The first equation is simply `x = 8y` . The second equation reduces to `0.25 - x = 10y` , which, using equation one, can be easily solved as: `0.25 - 8y = 10y rightarrow 0.25 = 18y` . This means that `y = 0.25/18 =0.013888889 hrs` , which is just 50 seconds.` `

Hence, the runners will meet after 50 seconds.

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To check, simply solve for the distance travelled by both after 50secs or 0.01389 hours.

Distance (A) = 8 mph * 0.01389 h = 0.11112 miles

Distance (B) = 10 mph * 0.01389 h = 0.1389 miles

Total (A+B) = 0.11112 + 0.1389 = 0.25 miles

The answer must be correct. The total distance is 0.25 miles, and hence the two must have met. Also, B travelled more distance since s/he is faster. If they had the same speed, they would simply meet at the center.