5. Linda goes to visit a friend who lives in Lompoc. It is 390 miles from Linda's house to her friend's house. When she came back, she didn't stop, so it took her a half an hour less time and her average speed was 5 miles per hour faster. How fast was she going when she went down to Lompoc, and how fast was she going when she drove back?

First we define our unknowns. In this case they are looking for how fast she was driving when she went down to Lompoc and how fast she was driving when she drove back. We could set this up with two unknowns, but since we know that she averaged 5 mi/hr faster coming back we could

let   x = how fast she was driving going down

and then we would have that

x + 5 = how fast she drove on the way back.

Your authors suggest we make up a   d = rt   table to organize our thinking.

Once we have two columns filled in we can fill in the third column by using the appropriate form of the   d = rt   equation. In this case we need to fill in the column for the time so we use the equation

t = d/r

Now the sentence "It took her a half hour less time to get back." can be translated as

This gives us a rational equation. Clear denominators. The smallest common denominator is   2x(x + 5)

Multiply the   2x(x + 5)   by both terms on the right side.

Cancel and assemble the surviving factors.

(2x)390 = 2(x + 5)390 = x(x + 5)

Remove parentheses and multiply.

780x = 780x + 3900 - x2 - 5x

After subtracting 780x from both sides, transpose all the terms to the left.

x2 + 5x - 3900 = 0

Will this factor? Are there factors of 3900 which have a difference of 5? Survey all the ways to multiply two numbers to get 3900.

Even with a rather large number like 3900, there are not that many ways to get tow numbers to multiply to it. We can run through all possibilities in not too much time. Start with 1, which will, of course, go into 3900, and after 1, try 2. It is important to try all of the possibilities, because you would hate to miss the one that worked.Try all of the numbers until you get to a number which is bigger than the last number you got in the column of quotients. At that point, if there is a bigger number that goes into it, its quotient will be smaller than the last number you tried, and we have considered all of those. We find that there is a pair that have a difference of   5,   viz.,   60 and 65.   So it will factor.

(x - 60)(x + 65) = 0

and set the factors = 0.

x - 60 = 0       x + 65 = 0

x = 60         x = -65

Many people will miss out on all of this fun by simply using the quadratic formula.

With the quadratic formula, one has to know the square root of 15625.

Either way, we get two answers. One is 60 mi/hr and the other one is -65 mi/hr. Of the two, 60mi/hr makes more sense. If she was driving to Lompoc at 60 mi/hr then it would take her 6.5 hours .If she came back averaging 5 mi/hr faster she would be going at 65 mi/hr, and then it would take her 6 hours which would be 1/2 hour less time, and the answer checks.

The answer of -65 mi/hr would take a little more imagination to check. It would mean that she was driving to Lompoc at 65 mi/hr backwards. If she were driving backwards then she would have been in Lompoc 6 hours ago. To get her speed coming back, adding 5 mi/hr to -65 mi/hr would give us a speed of -60 mi/hr, and it would then take her -6.5 hrs which is 1/2 less than -6 hours. However, most authors would have us reject such an answer as being meaningless, but how much fun is that?