The Power of Formulas

If you are over 45, you can remember problem like this from Algebra I.

                         Dave can do a job in 5 hours and his
                    friend Moreno can do it in 4 hours.
                    How long would it take if they did
                    it together?

Not only do I remember doing them, but when I began my career as a teacher in
1969 the same text was being used, and I found myself sending my class home
with those same problems.  However, realizing they were all the same, except
that they might have a twist, Dave got to work one hour late, or something.
I thought there must be a better way, and that is what I asked my class.

This is why.  Think of a circle.  Do we find the Area of one circle differently
then in another.   NO!  A circle is a circle.

We have a  FORMULA:  RT = 1

This means the RATE/HOUR x TIME WORKED = 1
                 1 MEANS THE ENTIRE JOB
  
 DAVE above does 1/5 of the job per hour because he does the job in 5 hours.

 MORENO does ¼ of the job per hour because he does the job in 4 hours.

             1/5 x 5 = 1       and         ¼ x 4 = 1

Now we are simply combining their work.   R₁T + R₂T =1

The 1 and 2 in the formula are subscripts, and are just used  as counters

As I stood before my classes that 1st year I said,

           "Lets do it in general, using variables"

     (1/a) (x) + (1/b)(x) = 1

Here a and b stand for the times each spent working alone, and x is the
combined timed.  Now we solve.

           x/a  +  x/b  =  1        Our common denominator is ab

           (bx   +  ax)/ab   =  1/1  Now use the rules of proportions.

           (bx  +  ax)  =  ab   Factor x out of both terms on left

           x (a + b) = ab  Divide by sum (a + b)

                      x  = ab/(a + b)


Now think of our friends Dave and Moreno
            Dave = 5 hours
            Moreno = 4 hours

  Thus if we use our new formula: x = (5)(4)/(5 + 4)

                                                       x = 20/9 hours or 2 and 2/9 hours

With the same Algebraic tools we can create a formula for one of them
being 1 hour late.  With these formulas we can solve a difficult Algebra I
problems as easily as we use A = bh for a Rectangle.

Add this formula to you list. But I suggest deriving it in an Algebra I
class can inspire students to feel better about story problems, and look for
new strategies to solve them.