Question 356064
<pre>
Let the 1st traveler's rate be x mph
Let the 2nd traveler's rate be y mph

Let's take a look at the moment when they met, which was 9:05.

At 9:05 the 1st traveler, who started at 5AM, had been traveling for
4 hours and 5 minutes, or {{{4&5/60}}} hours or {{{4&1/12}}} hours or {{{49/12}}} hours.  
At rate x mph, he had traveled {{{expr(49/12)x}}} or {{{49x/12}}} miles.

At 9:05 the 2nd traveler, who started at 7AM, had been traveling for
2 hours and 5 minutes, or {{{2&5/60}}} hours or {{{2&1/12}}} hours or {{{25/12}}} hours.  
At rate y mph, he had traveled {{{expr(25/12)y}}} or {{{25y/12}}} miles. 

When they met at 9:05, each had the same number of miles yet to go that 
the other had already been.  That is, the 1st traveler had {{{25y/12}}}
miles yet to go and the 2nd traveler had {{{49x/12}}} miles yet to go.

We are told that they arrived at the same time.  So we will calculate each
traveler's time after 9:05 and set them equal. 

Since {{{TIME=DISTANCE/RATE}}}, the 1st traveler's time was his distance yet
to go, {{{25y/12}}}, divided by his rate x, or {{{(25y/12)/x}}} or {{{(25y/12)(1/x)}}} or {{{(25y)/(12x)}}} hours till he 
reached town B.

Also the 2nd traveler's time was his distance yet to go, {{{49x/12}}}, divided
by his rate y, or {{{(49x/12)/y}}} or {{{(49x/12)(1/y)}}} or {{{(49x)/(12y)}}} hours till he reached town A.

Since they arrived at their respective destinations at the same time, we set
these times equal:

     {{{(25y)/(12x)}}}{{{""=""}}}{{{(49x)/(12y)}}}

Multiplying both sides by 12 eliminates the 12's from the denominators:

     {{{(25y)/x}}}{{{""=""}}}{{{(49x)/y}}}

Cross multiplying:

     {{{25y^2}}}{{{""=""}}}{{{49x^2}}}

Taking positive square roots of both sides

            {{{5y}}}{{{""=""}}}{{{7x}}}

Dividing both sides by 5x

            {{{y/x}}}{{{""=""}}}{{{7/5}}}


The 1st traveler's time to reach his destination after 9:05 was

{{{(25y)/(12x)}}} which equals {{{expr(25/12)*expr(y/x)}}} which equals 
      {{{5}}}
{{{expr(25/12)*expr(7/5)}}}{{{""=""}}}{{{expr(cross(25)/12)*expr(7/cross(5))}}}{{{""=""}}}{{{expr(5/12)*expr(7/1)}}}{{{""=""}}}{{{35/12}}}{{{""=""}}}{{{2&11/12}}}{{{""=""}}}{{{2&55/60}}} or 2 hours and 55 minutes after 9:05.
So the 1st traveler reached his destination at 12 noon.

It isn't necessary, but, as a check, let's see if the 2nd traveler also reached
his destination at 12 noon.

The 2nd traveler's time to reach his destination after 9:05 was

{{{(49x)/(12y)}}} which equals {{{expr(49/12)*expr(x/y)}}} which equals 
      {{{7}}}
{{{expr(49/12)*expr(5/7)}}}{{{""=""}}}{{{expr(cross(49)/12)*expr(5/cross(7))}}}{{{""=""}}}{{{expr(7/12)*expr(5/1)}}}{{{""=""}}}{{{35/12}}}{{{""=""}}}{{{2&11/12}}}{{{""=""}}}{{{2&55/60}}} or 2 hours and 55 minutes after 9:05.
So the 2nd traveler also reached his destination at 12 noon.

So the 1st traveler started at 5AM and arrived at noon, so his time was 7 hours.
The 2nd traveler started at 7AM and arrived at noon, so his time was 5 hours.

[Notice that this problem is independent of their speeds and the distance they
traveled.]

Edwin</pre>