Question 52337
<pre><font size = 5><b> 
The Question: 
A train traveled 960 miles from Town A to Town B.
If the train had traveled from Town A to Town B 
at an average speed of 12 miles per hour less, 
it would have taken 4 more hours. 
What was the train's average speed during that 
journey? 

The explanations in some of those online courses are 
not so good. Here's the way I would solve it:

Make this chart:
 
                    DISTANCE   RATE       TIME
  ACTUAL TRIP                                   
HYPOTHETICAL TRIP                                  

Let the speed on the actual trip be x.  Fill that in 
for the RATE of the ACTUAL TRIP.

                    DISTANCE   RATE       TIME
  ACTUAL TRIP                   x              
HYPOTHETICAL TRIP                               

>>...If the train had traveled from Town A to Town B 
at an average speed of 12 miles per hour less...<<

So we subtract 12 from x, getting x-12 and fill that 
in for the RATE for the HYPOTHETICAL TRIP.

                    DISTANCE   RATE       TIME
  ACTUAL TRIP                   x              
HYPOTHETICAL TRIP              x-12                

We are told that the DISTANCE for both the ACTUAL 
and the HYPOTHETICAL trip is 960 miles, so we fill 
that in:

                    DISTANCE   RATE       TIME
  ACTUAL TRIP          960      x              
HYPOTHETICAL TRIP      960     x-12              

Now we fill in the TIMEs using TIME = DISTANCE/RATE

                    DISTANCE   RATE       TIME
  ACTUAL TRIP          960      x        960/x 
HYPOTHETICAL TRIP      960     x-12    960/(x-12)

Now we look for what we haven't used:

>>...it [THE HYPOTHETICAL TRIP] would have taken 4 
more hours...<<

So  

HYPOTHETICAL TRIP TIME = ACTUAL TRIP TIME + 4 HOURS

or

960/(x-12) = 960/x + 4

Divide thru by 4

240/(x-12) = 240/x + 1

Multiply through by LCD = x(x-12)

240x = 240(x-12) + x(x-12)

240x = 240x - 2880 + x² - 12x

   0 = x² - 12x - 2880

Solve this by the quadratic formula

        x² - 12x - 2880 = 0

The quadratic formula is:
                  ______ 
            -b ± <font face = "symbol">Ö</font>b²-4ac
        x = —————————————
                2a 

where a = 1; b = -12; c = -2880

                       __________________ 
             -(-12) ± <font face = "symbol">Ö</font>(-12)²-4(1)(-2880)
        x = ——————————————————————————
                     2(1) 
                   _________ 
             12 ± <font face = "symbol">Ö</font>144+11520
        x = ——————————————————
                   2
                   _____ 
             12 ± <font face = "symbol">Ö</font>11664
        x = ————————————————
                  2
                    
             12 ± 108
        x = ——————————
                2  

Using the +

             12 + 108
        x = ——————————
                2

             120
        x = ————— = 60 mph
              2

Using the -

             12 - 108
        x = ——————————
                2

             -96
        x = ————— = -48 mph
              2

so we discard that and the only answer is 60 mph.

Edwin</pre>