<PRE><B>Question 1125197</B>
&lt;br&gt;
Let the speed of the train be x (mph).  Then the problem tells us that the time required to travel 1650 miles at speed x is 5 hours longer than if the train traveled at a speed of x+11:&lt;br&gt;
{{{1650/x = 1650/(x+11)+5}}}&lt;br&gt;
Multiply everything by the common denominator to clear the fractions:&lt;br&gt;
{{{1650(x+11) = 1650(x)+5(x)(x+11)}}}
{{{1650x+18150 = 1650x+5x^2+55x}}}
{{{0 = 5x^2+55x-18150}}}
{{{0 = x^2+11x-3630}}}&lt;br&gt;
To solve that by factoring, you would need to find two numbers whose difference is 11 and whose product is 3630.  That would be quite a task for most people....&lt;br&gt;
Of course you could plug the coefficients into the quadratic formula; and in fact if you are good with mental arithmetic, that turns out not to be too difficult.&lt;br&gt;
But, since using formal algebra leads us down a path where there is a lot of work to do, let&#39;s see if we can get to the answer with less work, without the formal algebra -- by looking at the original equation and doing some logical reasoning.&lt;br&gt;
{{{1650/x = 1650/(x+11)+5}}}&lt;br&gt;
Note that we can divide the whole equation by 5 to get an equivalent equation with smaller numbers:&lt;br&gt;
{{{330/x = 330/(x+11)+1}}}&lt;br&gt;
This equation says that 330 divided by some number is a whole number, and 330 divided by a number that is 11 larger is also a whole number; and the difference between those two whole numbers is 1.&lt;br&gt;
So look at the integer factors of 330 and find two of them that differ by 11, and dividing 330 by those factors gives two other integers that differ by 1.&lt;br&gt;&lt;pre&gt;
  1 * 330
  2 * 165
  3 * 110
  5 *  66
  6 *  55&lt;/pre&gt;
AHA!  There it is!  330 divided by 55 and by 66 produces two integers that differ by 1.&lt;br&gt;
And so 1650 divided by 55 and by 66 produces two integers that differ by 5, as required:&lt;br&gt;
1650 miles at 55 mph = 1650/55 = 30 hours
1650 miles at 66 mph = 1650/66 = 25 hours -- which is 5 hours less than 30 hours.&lt;br&gt;
ANSWER:  The speed of the train is 55mph.</PRE>