SOLUTION: A train covered 1 comma 650 1,650 mi at a certain speed. Had the train been able to travel 11 11 mph​ faster, the trip would have been 5 5 hr shorter. How fast did t

Algebra ->  Expressions-with-variables -> SOLUTION: A train covered 1 comma 650 1,650 mi at a certain speed. Had the train been able to travel 11 11 mph​ faster, the trip would have been 5 5 hr shorter. How fast did t      Log On


   

Question 1125197: A train covered
1 comma 650
1,650 mi at a certain speed. Had the train been able to travel
11
11 mph​ faster, the trip would have been
5
5 hr shorter. How fast did the train​ go
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


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:

1650%2Fx+=+1650%2F%28x%2B11%29%2B5

Multiply everything by the common denominator to clear the fractions:

1650%28x%2B11%29+=+1650%28x%29%2B5%28x%29%28x%2B11%29
1650x%2B18150+=+1650x%2B5x%5E2%2B55x
0+=+5x%5E2%2B55x-18150
0+=+x%5E2%2B11x-3630

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....

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.

But, since using formal algebra leads us down a path where there is a lot of work to do, let'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.

1650%2Fx+=+1650%2F%28x%2B11%29%2B5

Note that we can divide the whole equation by 5 to get an equivalent equation with smaller numbers:

330%2Fx+=+330%2F%28x%2B11%29%2B1

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.

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.
  1 * 330
  2 * 165
  3 * 110
  5 *  66
  6 *  55

AHA! There it is! 330 divided by 55 and by 66 produces two integers that differ by 1.

And so 1650 divided by 55 and by 66 produces two integers that differ by 5, as required:

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.

ANSWER: The speed of the train is 55mph.