SOLUTION: Justin recently drove to visit his parents who live 504 miles away. On his way there his average speed was 16 miles per hour faster than on his way home (he ran into some bad weath

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Question 1182960: Justin recently drove to visit his parents who live 504 miles away. On his way there his average speed was 16 miles per hour faster than on his way home (he ran into some bad weather). If Justin spent a total of 16 hours driving, find the two rates.
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52797)   (Show Source): You can put this solution on YOUR website!
.
Justin recently drove to visit his parents who live 504 miles away.
On his way there his average speed was 16 miles per hour faster than on his way home (he ran into some bad weather).
If Justin spent a total of 16 hours driving, find the two rates.
~~~~~~~~~~~~~~~~~~ 


Let x be the slower rate, in miles per hour.

Then the faster rate is  (x+16) mph.


The time traveling "there" is    hours.

The time traveling  back   is         hours.


The total time is 16 hours, giving you the "time" equation


     +  = 16  hours.


To solve this equation, multiply both sides by x*(x+16) and simplify


    504x + 504(x+16) = 16x*(x+16)

     63x + 63(x+16) = 2x*(x+16)

     63x + 63x + 63*16 = 2x^2 + 32x

      2x^2 - 94x - 1008 = 0

       x^2 - 47x - 504 = 0


You can solve it by using the quadratic formula or by factoring


      (x-56)*(x+9) = 0


Since you need the positive root, only, you choose  x = 56.


ANSWER.  The rate traveling back is 56 mph;  the rate traveling to "there" was 56+16 = 72 mph.


CHECK.   +  = 7 + 9 = 16  hours.   ! correct !

Solved.



Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


Of course you should understand the algebraic solution to the problem, as shown by the other tutor.

However, if a formal algebraic solution is not required, and the speed of solving the problem is important -- as in a timed math competition -- you can find the answer by playing with numbers.

There are several pairs of numbers -- hours and speed in mph -- that give a product of a distance of 504 miles. Search for two such pairs in which the sum of the two hour numbers is 16 and the difference between the two speed numbers is 16.

Ignore unreasonable pairs, such as 2 hours and 252mph....
  6 * 84
  7 * 72
  8 * 63
  9 * 56
 12 * 42

The two pairs that satisfy the conditions of the problem are 7*72 and 9*56. The sum of the two hours numbers is 16, and the difference between the two speed numbers is 16.

ANSWER: The two rates are 72 and 56 mph.
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