Question 1163467
<pre>
I actually see nothing wrong with or unprofessional about the problem as
stated.

Ikleyn assumes all students who post here are from English-speaking
countries.  It doesn't occur to her that often problems are the
student's own translations from other languages.  So if a problem is not
stated in the best English, she will complain.  Don't take it personally.

But, as I said above, I think the wording is OK.

What she didn't notice was that it's easy to tell which driver is 
faster and thus which one will thus drive 40 km farther than halfway,
and which one is slower and will drive 40 km less than halfway. 

Let x = distance from A to B (or B to A). 
</pre>
Sam drives from City A to City B, and he takes 8 hours for the entire trip.
Richard drives from City B to City A, and he takes 10 hours for the trip
assuming they are driving on the same road.
<pre>
Make this chart, using rate = distance/time

        |  distance  | rate  | time  |
--------------------------------------
Sam     |      x     |  x/8  |   8   |        
Richard |      x     | x/10  |  10   |

Sam's rate is x/8 while Richard's rate is x/10.
</pre>
If Sam leaves City A and Richard leaves City B at the same time, how many
hours will they meet at a place 40 km away from the halfway? 
<pre>
Let t = the time at which they meet.

Sam is faster, so he will travel 40 km farther than halfway, which is
1/2x+40. 

Richard is slower, so he will travel 40 km less than halfway, which is
1/2x-40.  Add those two lines onto the chart:

        |  distance  | rate  | time  |
--------------------------------------
Sam     |      x     |  x/8  |   8   |        
Richard |      x     | x/10  |  10   |
Sam     | (1/2)x+40  |  x/8  |   t   |
Richard | (1/2)x-40  | x/10  |   t   |

Use distance = rate × time to form two equations:

{{{system(expr(1/2)x+40=expr(x/8)t,expr(1/2)x-40=expr(x/10)t)}}}

Multiply the first equation through by 8 and the second one
through by 10:

{{{system(4x+320=xt, 5x-400=xt)}}}

Since xt = xt, we have the equation:

4x + 320 = 5x - 400 
     720 = x         

So the distance from A to B is 720 km.

Substitute 720 for x in:

{{{4x+320=xt}}}

{{{4(720)+320=(720)t}}}

{{{2880+320=720t}}}

{{{3200=720t}}}

{{{3200/720=t}}}

{{{40/9=t}}}

{{{matrix(1,2,4&4/9,hours)}}}

That's the answer as a mixed number.

Edwin</pre>