document.write( "Question 1198566: A train traveling from Avondale to Bimmaeus stops at a cracked rail after 1 hour, and the engineer spends half an hour fixing it. After which it proceeds at four fifths its usual rate to make it easier to check for problems, arriving at Bimmaeus 2 hours late. If the train had covered 80 miles more before the accident, it would have been just one hour late. What is the train's usual speed? \n" ); document.write( "

Algebra.Com's Answer #832209 by Edwin McCravy(20055) $\"\"$ $\"About$

You can put this solution on YOUR website!

 \r\n" );
document.write( "Ikleyn's solution is CORRECT.  I overlooked the word \"MORE\", and solved this\r\n" );
document.write( "problem instead, where the \".....\" is below.  It is a much harder problem that\r\n" );
document.write( "way.

A train traveling from Avondale to Bimmaeus stops at a cracked rail after 1
\n" ); document.write( "hour, and the engineer spends half an hour fixing it. After which it proceeds at
\n" ); document.write( "four fifths its usual rate to make it easier to check for problems, arriving at
\n" ); document.write( "Bimmaeus 2 hours late. If the train had covered 80 miles.....before the
\n" ); document.write( "accident, it would have been just one hour late. What is the train's usual
\n" ); document.write( "speed?

Here is the more difficult solution to the problem without the word \"MORE\": \r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "The above drawing represents the two situations.\r\n" );
document.write( "\r\n" );
document.write( "x = distance from A to cracked rail\r\n" );
document.write( "y = distance from cracked rail to B\r\n" );
document.write( "z = distance from cracked rail to B if cracked rail were 80 miles from A.\r\n" );
document.write( "\r\n" );
document.write( "Let r = the usual speed of the train\r\n" );
document.write( "Let h = the number of hours the train would take to go from A to B at \r\n" );
document.write( "the usual speed r.  So using d=rt, we have AB = hr. From that and the \r\n" );
document.write( "drawing above, we have:\r\n" );
document.write( "\r\n" );
document.write( "(1)  Distance AB = x+y = 80+z = hr\r\n" );
document.write( "\r\n" );
document.write( "Make this DRT chart for the 4 distances in\r\n" );
document.write( "the drawings above:\r\n" );
document.write( "\r\n" );
document.write( "Distance | Rate | time\r\n" );
document.write( "    x    |  r   |  1\r\n" );
document.write( "    y    | 4/5r | (h+2)-(1)-(1/2) = h+1/2\r\n" );
document.write( "   80    |  r   | 80/r\r\n" );
document.write( "    z    | 4/5r | (h+1)-(80/r)-(1/2) = h-80/r+1/2\r\n" );
document.write( "\r\n" );
document.write( "I will explain the times for distances y and z.\r\n" );
document.write( "\r\n" );
document.write( "Time for traveling the y miles:\r\n" );
document.write( "The train was 2 hours late, so it arrived in h+2 hours\r\n" );
document.write( "after leaving A. However, 1 hour was spent traveling \r\n" );
document.write( "the x miles, and another 1/2 hour was spent fixing the \r\n" );
document.write( "cracked rail.  So we cannot count those times for the \r\n" );
document.write( "time traveling the y miles. So we must subtract those \r\n" );
document.write( "times from the total h+2 hours. Thus the time for \r\n" );
document.write( "traveling the y miles is\r\n" );
document.write( "(h+2)-(1)-(1/2) = h+1/2\r\n" );
document.write( "\r\n" );
document.write( "Time for traveling the z miles:\r\n" );
document.write( "The train would have been 1 hour late, so it would have \r\n" );
document.write( "arrived in h+1 hours after leaving A.  However, using \r\n" );
document.write( "t=d/r, 80/r hours would have been spent traveling the \r\n" );
document.write( "80 miles, and 1/2 hour would have been spent fixing the \r\n" );
document.write( "cracked rail.  So we cannot count those times \r\n" );
document.write( "for the time traveling the z miles. So we must subtract \r\n" );
document.write( "those times from the total h+1 hours.  Thus the time for \r\n" );
document.write( "traveling the z miles is\r\n" );
document.write( "(h+1)-(80/r)-(1/2) = h-80/r+1/2.\r\n" );
document.write( "\r\n" );
document.write( "So from the drawing above, we have distance from A to B:\r\n" );
document.write( "\r\n" );
document.write( "using (1) above and using d = rt in the chart above, we \r\n" );
document.write( "have this system:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Since the 2nd equation is x = r, we replace x by r, and\r\n" );
document.write( "simplify the other equations:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "From the 1st equation we replace hr by r+y in the 2nd and 3rd equation,\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "They simplify to\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "From the 2nd equation, substituting 6r for y in the other two equations\r\n" );
document.write( "gives \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Divide the 2nd equation through by 5\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Substituting 6r-64 for z in the first equation.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "   <-- answer\r\n" );
document.write( "\r\n" );
document.write( "It isn't required, but if we get all the other variables,\r\n" );
document.write( "h=7 hours, x = 16 miles, y=96 miles,\r\n" );
document.write( "z=32 miles, so the distance from A to B is 112 miles.\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "Edwin

\n" ); document.write( "

\n" );