Finding Sally's Hourly Wage

Date: Mon, 7 Nov 1994 17:37:53 -0500
From: Rich Joseph
Subject: Ashley Petrylak

Dear Dr. Math,

My name is Ashley Petrylak and I'm In the MathCounts program at 
E.T.Richardson Middle School.I need to do two worksheets for 
homework. I am having trouble with the following problems.

Sally works at a grocery store.  In one week she earned $390 for 
working 47 hours, of which 7 hours were overtime.The next week 
she earned $416 for working 50 hours, of which 8 hours were overtime.
What is Sally's hourly rate?

Here's what I've already tried:

40r + (7r + 7o) = $390
47r + 7o = $390

42r + (8r + 8o) = $416
50r + 8o = $416

When I tried to get r by itself, I had trouble because I couldn't make sense of
the division, 47r + 7o
              ________
                 47
Please help!

Date: Mon, 7 Nov 1994 18:22:56 -0500 (EST)
From: "Michael W. S. Morton"
Subject: Re: Ashley Petrylak

Great Problem!
 
The first thing I did when I looked at the problem is the same thing you 
did, only a little different.  Let's define Sally's hourly rate and her 
overtime rate.  Say,

        Sally's hourly rate = r
        Sally's overtime rate = o

Then we get the two equations:

40r + 7o = $390 since the first 40 hours were normal pay, and the next 7 
                hours were overtime pay.
42r + 8o = $416 again, the first 42 hours were normal pay, the next 8 
                overtime.

So, we need to solve for r which is possible because we have two 
equations and two unknowns (r and o).

Let's try multiplying the first equation by 8 and the second equation by -7.

 8 * (40r + 7o = $390) =>  320r + 56o =  $3120
-7 * (42r + 8o = $416) => -294r - 56o = -$2912

Then, let's add the two equations above together:

         320r + 56o =  $3120
        -294r - 56o = -$2912
        --------------------
          26r  + 0o  = $ 208

Now we have one equation and only one unknown!  See if you can solve 
that, and then you could even find Sally's overtime pay by substituting 
that value back into one of the other earlier equations!  Hope all that 
makes sense.... write us back if it doesn't!

                                                -MORTON, Doctor on call.
                                                X-Mailer: Eudora 1.3.1+Swarthmore

Date: Mon, 7 Nov 1994 18:30:06 +0900
Subject: Re: Ashley Petrylak

Hi Ashley-

        Thanks for writing to us. I did MathCounts when I was in 7th and
8th grade myself. So I am very excited to hear from you.

You have set up the problem perfectly. In order to isolate the r, you need
to subtract the 7o from both sides first and then divide by 47.

So you get:

47r + 7o = $390  is the same as  47r = 390 - 7o

          which  is the same as        390 - 7o
                                   r = --------
                                          47

Then take this value for r and plug it into your second equation (50r+8o=516)
Then you will have all o's, so solve for o and your almost there.

I hope this helps.  Please write back if its not clear or if you have any
other questions.

-Margaret, a Math Doctor