Employee Pay Schedule

Date: 07/12/2003 at 13:51:36
From: Tom 
Subject: How do I figure out a pay schedule for my employees?

My employees get paid piece rate on a job:

  The job pays $500.00
  Moe worked    8.5 hrs @ 40% of gross pay
  Larry worked 11.5 hrs @ 30% of gross pay
  Curly worked 2.75 hrs @ 30% of gross pay

I can't seem to figure out a math formula for them when they work 
different numbers of hours on the job. If you could show me the
formula for this I would be very appreciative.

Date: 07/24/2003 at 16:43:42
From: Doctor Paul
Subject: Re: how do I figure out a pay schedule for my employees?

Hi Tom,

Let's look at an easier problem first. Suppose we had the same $500 
check to be divided among Larry, Moe, and Curly for work they did on 
one particular day. Larry and Moe get 40% of the gross pay and Curly 
(who is apparently new on the job) gets paid less - 20% of the gross 
pay. If they all worked the same 8-hour day, then we would split the 
money as follows:

  Larry gets 40% of $500 = $200
  Moe   gets 40% of $500 = $200
  Curly gets 20% of $500 = $100

This is easy. Let's vary the situation a bit:

Suppose the same pay scale, but imagine that Larry got sick after 
lunch (he had some bad seafood) so he only worked four hours. Moe and 
Curly stayed behind and finished the job. How would we divide the 
money in this scenario? Here is my thinking:

For the four hours in the morning that the three worked together, the 
money should be divided as was the case the previous day. The four 
hours represents one half of the day so the money to be divided should 
be one half of the gross: $250.

So for the first half of the day, we have:

  Larry gets 40% of $250 = $100
  Moe   gets 40% of $250 = $100
  Curly gets 20% of $250 =  $50

For the second half of the day, Larry didn't work so he shouldn't get 
paid anything. Instead, Moe and Curly need to somehow share the 
remaining $250. But how should it be divided? Moe usually gets 40% of 
the money and Curly usually gets 20% of the money. But that's only 60% 
of the money. This obviously isn't right. We want to divide the money 
in such a way that:

  1.  All of the money is gone
  2.  The ratio of money earned remains the same.

The idea here is that when we are done dividing the $250, Moe will get 
twice as much as Curly since he makes twice as much as Curly does (40% 
of gross is twice as much as 20% of gross).

Above, I noted that if we paid Moe 40% and Curly 20% then we would 
only have spent 60% (or 6/10) of the money. In order to keep the ratio 
of money earned the same, we need to scale this so that when we sum 
the percentages, we have spent 100% of the money. The way to do this 
is to multiply both of the percentages by the reciprocal of 6/10 - 
namely 10/6.

  40% * 10/6 = 66.6667%
  20% * 10/6 = 33.3333%

This works perfectly. Moe will get twice as much as Curly and we will 
have spent 100% of the money. So for the second half of the day, we 
have:

  Moe   gets 66.6667% of $250 = $166.67
  Curly gets 33.3333% of $250 =  $83.33

This brings our daily total to:

  Larry: $100
  Moe:   $266.67
  Curly: $133.33

I think you'll agree that this was a fair way to divide the money in 
this scenario.

Now let's look at your scenario:

  Moe   worked  8.5 hrs @ 40% of gross pay
  Larry worked 11.5 hrs @ 30% of gross pay
  Curly worked 2.75 hrs @ 30% of gross pay

Basically, I see this as as 11.5-hour day where the crew worked as 
follows:

The group worked for 2.75 hours together as a trio, at which point 
Curly went home. For the next 5.75 hours, Moe and Larry worked 
together. Then Moe went home. Larry stayed behind for the final three 
hours to finish the project.

Obviously, your crew might not have worked the hours in this order.  
But for payment purposes, the order in which they choose to work their 
hours is unimportant.

2.75 hours out of an 11.5-hour day represents 23.913% of the day.  
Now, 23.913% of the $500 gross is $119.56, to be split as follows:

  Moe   gets 40% of $119.56 = $47.82
  Larry gets 30% of $119.56 = $35.87
  Curly gets 30% of $119.56 = $35.87

The next segment of the day is 5.75 hours - 50% of 11.5. So 50% of the 
$500 gross - namely $250 - needs to be divided between Moe and Larry.  
In doing this dividing, we want to make sure that the ratio of money 
earned remains the same.  

Moe earns 4/3 as much as Larry (40%/30% = 4/3). If all three people
were present, the earnings of Moe and Larry would represent 70% (or
7/10) of the total. Since they are now 100% of the workforce, we need
to scale their payment amounts appropriately. We do this by 
multiplying by the reciprocal of 7/10 -  namely 10/7.

  10/7 * 40% = 57.143%
  10/7 * 30% = 42.857%

You can verify that the ratio of earnings (4 to 3) has been preserved.  
So 

  Moe   will get 57.143% of $250 = $142.86
  Larry will get 42.857% of $250 = $107.14

Finally, for the last three hours Larry worked by himself, so he'll 
get 100% of the money. Three hours represents 26.086% of the day.  
So 

  Larry will get 26.086% of $500 = $130.43

To summarize:

  Moe   gets: $47.82 + $142.86           = $190.68
  Larry gets: $35.87 + $107.14 + $130.43 = $273.44
  Curly gets: $35.87                     =  $35.87

The total is: $190.68 + $273.44 + $35.87 = $499.99

It looks as if we have a rounding error. We'll let Larry, Moe, and 
Curly argue over the final penny. I'm sure it'll make for hours of 
family entertainment.  :-)

I hope this helps.  Please write back if you'd like to talk about 
this some more.

- Doctor Paul, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 08/01/2003 at 15:29:27
From: Doctor Ian
Subject: Re: How do I figure out a pay schedule for my employees?

Hi Tom, 

In addition to the way that Dr. Paul approached the problem, I'd like
to propose a second way to look at it. 

To make the numbers simpler, I'm going to assume that Curly worked 4
hours, instead of 2.75 hours.  This will give me a slightly different
outcome than you want, but it will make the method clearer. 

Suppose that before the job starts, we hand out the money, assuming
that all three workers will work the same number of hours.  That is,

  Moe   gets 40% of $500, or $200
  Larry gets 30% of $500, or $150
  Curly gets 30% of $500, or $150

Of course, they _don't_ end up working the same number of hours.

  Moe works    8.5 hours
  Larry works 11.5 hours
  Curly works  4.0 hours          (in this version)

That's a total of 24 hours; so each person _should_ have worked for 
8 hours. Now, what does this mean? It means that Moe worked 1/2 of one
of Curly's hours; and Larry worked 3.5 of Curly's hours.  

Half an hour is 1/16 of Curly's expected hours; so he should give Moe
1/16 of the money he was paid, or $150/16.  

Similarly, 3.5 hours is 7/16 of Curly's hours; so he should give Larry
7/16 of the money he was paid, or $150*(7/16).  

In essence, you're staying out of it, and letting Curly subcontract
his part of the job.

Does this make sense? 

Now let's try it with the actual numbers.  The total number of hours 
is 

  8.5 + 11.5 + 2.75 = 22.75

so each person would be expected to work 

  22.75 / 3 = 7.58 hours

Now, Curly only works 2.75 hours, which means he's going to 
subcontract out 

  7.58 - 2.75 = 4.83

of his hours. The amount picked up by Moe is 

  8.5 - 7.58 = 0.92 hours

and the amount picked up by Larry is

  11.5 - 7.58 = 3.92 hours

(Note that because we're using decimal approximations instead of exact
fractions, these add up to 4.84 hours. We'll make Curly eat the extra
0.01 hour.)

Anyway, Moe should get 

  (0.92/7.58)*$150 = $18.21

from Curly, and Larry should get

  (3.92/7.58)*$150 = $77.57

from Curly.  That leaves Curly with 

  $150 - ($18.21 + $77.57) = $54.22

Moe ends up with 

  $200 + $18.21 = $218.21

and Larry ends up with 

  $150 + $77.57 = $227.57

which all adds up to $500, as it should. 

Note that Dr. Paul and I came up with different answers. What
conclusions can we draw from that? 

The most important conclusion, I think, is that math can only do so
much for you in solving a problem. Often 'doing the math' is
relatively easy compared to the tougher job of figuring out how to
interpret the situation that gives rise to the problem.  We've shown
you two different ways to interpret your situation.  There may be
others as well.  In the end, you have to choose one and go with it.

I hope this helps! 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/