Little help with labor percentage and how to set up an equation

My girlfriend is the co-manager at a restaurant and texted me today at work asking for my help to figure out what percentage of labor she will need to run today to reach a 16% average for the week considering that before the day started, the percent for the week was at 17.42 This was my response to her in a couple of emails:

17.42%(85.7) + X%(14.3) = 16%

Where 17.42% is the % of labor for the week thus far
where 85.7 is the % of days in which the labor has been averaged so far (6/7)
where X is the unknown % needed to yield 16% labor at EOW
where 14.3 is the % of days in which labor has not yet been averaged (1/7)
where 16% is obviously the desired labor % for EOW

SOLVE FOR X

1493 + 14.3X = 1600

where 1493 is calculated by multiplying WTD labor with average labor thus far
where 14.3X is the remaining % of labor to be calculated multiplied by the unknown, (X)
where 1600 is the % labor we want to reach multiplied by 100 (because we want all of the days of labor during the week accounted for in our percentage/average aka 100%)

CONTINUE TO SOLVE FOR X

14.3X = 107

where 14.3X is left alone for the time being
where 107 is calculated by subtracting like terms to get X by itself (i.e 1600 - 1493 = 107)

CONTINUE TO SOLVE FOR X

x = 7.48

where 7.48 is the result of dividing 14.3 from both sides cancelling out the 14.3 on the left leaving 1X or just X = (14.3/107)

7.48 % labor is what is needed on Sunday to reach 16% total averaged labor for the week.

Another way of solving this with a trial-and-error type method is as follows:

17.42 x 6 = 104.52

where again 17.42 is the average % of labor for the week thus far
where 6 is the total number of days averaged into the weekly labor thus far
where 104.52 is the product of the two


So we have to find a product (104.52) that satisfies the condition that for 7 days the average will drop back down to 16% overall. So we'll have to try a few numbers and then divide by 7 appropriately. Why 7? Because 104.52 / 6 = 17.42 and the 6 represents the days of the week that have been figured so far and not the 7th. so let's try a few numbers shall we? We're going to look for a product that divided by 7 will be close to 16%.

Hmmm, let's start out with 120 as our product for the whole 7 day work week. Meaning that since Mon-Sat the product(the average labor % multiplied by days work thus far 6) was 104.52, than the labor % for Sunday would have to be 15.48 or (120 - 104.52)

So now we will divide 120 by 7 getting an average of 17.14% for the week. Seems reasonable right? Started Sunday with 17.42% and ending the day with a 15.48% labor would drop you down only .28% If it seems strange that you only went down by a little over a quarter of a percentage point after having your second best day in terms of labor for the week, it's because that one day(even both Monday and Sunday) are only 2/7th of your overall average and the other 5/7th of the average is well above 16%.


15.8% labor for one day is pretty reasonable but it still doesn't get you close to what you want to finish the week with so we'll try something a little more unlikely such as 7.48% labor for one day. What we do now is add that 7.48 to the original product of 104.52 and get 112. And once we have 112 we can divide that by the 7 days in the week getting 7/112 = 16% BY GOLLY WE FOUND IT!!! Unfortunately having a 7.48% labor for any day of the week let alone the slowest and most overtime ridden day of the week is next to impossible. You can also calculate for whatever percentage you want using this method it just may take some time to fudge the numbers to get exactly what you need or at least close to it. The first method is the most efficient of the two but neither are ideal. I know there is an easier way to set up an equation to find any value for X(labor needed) to get any other value of say Y(labor wanted) but I can't think of it for the life of me right now. Hope that helps my dear. I had fun doing this, I hope I didn't rack your brain too much lol

That was the first message I sent to which she was somewhat confused and replied that she added all of the totals of the labors for each day and then
added a projected 14% labor for today and divided by 7 getting what she said was "close to 16%" to which I replied:

Say you ended Sunday with a 13.48% labor. We'll add that to the 6 day product of 104.52 to get 118. 7/118 is still only 16.86% labor. Better but still not at or under 16%

Let's try 12.48 making it 7/117. even that is only 16.71% labor for the week. It's not until you get all the way down to 7.48 that you get to exactly 16% and not even under it. Sorry deary. Who knows, I could be completely wrong!!

Maybe I should have posted this next email before the last example I just gave so as to show how she and I got the 104.52 but I guess now will make it clear, so here is the final message I sent:

I also wanted to do what you did to show you that you either made a mistake in your calculation or just use the term "close to 16" very loosely, lol.

14.77
+17.67
+17.06
+18.08
+18.40
+18.47
+14.00 <==== Sunday's projected/hopeful labor

= 118.45 all 7 labors added together

/ 7 divided by the 7 days of the week giving us an average for the week of

16.92%

So if that's what you got then you did it correctly and just may have thought that since it was in the 16 range then anything under 14 for the day would get closer and closer to 16%. Which is true but at a very very slow rate. Instead of 14 for today let's say your sexy self runs a 10% labor for the day. Subtracting 4 from the total 7 day labor total we get 114.45 And once we divide that by the 7 days. we get 16.35 A lot closer to that 16% you're looking for, but you know as well as I do how difficult it is to drop 4% in labor on any given day. So again let's say you are just getting slammed all night and you only have 4 crew members and you are awesome enough to run a 7.48% labor for today. We'll take the total (118.45) minus 6.52(the difference between 14 and 7.48) and come up with 111.93. We'll again divide that by the 7 day work week and come with a total labor average for the week of...you guessed it...15.99% MUAH!!!

So what I'm really asking is for some help on creating an equation or an easier method than the ones I used to help find a certain % needed on a given day to satisfy a certain % average for the week. I'm almost positive that I came to the correct conclusion that only running a labor of 7.58 maybe 7.59 or lower is the only way that for the week, she would end at 16% labor or lower, the goal for the store. In trying to come up with an equation(keeping in mind I've been out of school for about 6 years now and haven't dealt with much algebra in quite some time) it reminded me somewhat of highschool where at the end of each semester each teacher would give the students an equation that would determine what we needed to get on the final to get a desired grade in the class based on the how the tests, quizzes, homework etc. were weighted but for the life of me can't find or remember it. I did find some calculators online that did the equation for you after plugging in certain criteria and I did come up with 7.48 or 7.49 and in one case 7.43, but I couldn't find the actual equation that is used. So if anyone can help it would be greatly appreciated and if I made any mistakes in any of my 3 original emails please feel free to let me know. I take criticism well! Also I'm pretty sure there is a way to figure out what you would need over a two day period(say it's the end of the night Friday and we are currently at a 17% labor and we want to see what we need together or individually Saturday and Sunday to get to 16%. Or say you figure that you have the first 5 days of the week and you figure based on projections that you'll finish Saturday with a 15.82% labor and want to know what you would need Sunday if and only if Saturdays projections are spot on.) Hell maybe I did ever1ything perfectly I dunno. I'm just bored and math is enjoyable for me. I used to be very good at it back in my high school days but since then I haven't had much use for it since I skipped college and went straight into the work force. If only I could go back, hehe