My family's bakeshop used to have three branches. I mention this because, until recently, the third branch ran for almost thirty years beside an auto-mechanic's garage somewhere on the edge of the Sta. Mesa district of Manila.

Last December, however, the new owner of the lot decided that they would no longer renew our rental... so, sometime around the first week of January, we finally shut down the place. In retrospect, this made for some interesting timing, as my mother had been wondering how to adjust the direction of her operations for some time now.

At the moment, the bakeshop is in negotiations for a new concessionaire in the heart of one of the local malls. Normally this would be quite simple — we'd just go to the mall owners, work out a rental arrangement, then set up shop on a given date. Instead, we've found out that the concept of "fixed rent" is apparently a thing of the past: Rather than pay a fixed amount per month, we're now being asked to provide a certain percentage of our gross sales (in addition to the standard utlities payment) in exchange for the space.

This brought up some interesting questions for my mother: Was this a good deal? Would the bakeshop be able to absorb a proportional share of its sales, rather than a given fixed amount? And if the offer was open to negotiation, for what percentage value should they negotiate?

And, because I was the family mathematician, all this landed on my table one evening sometime in the middle of last week. The worse part was that it was hard to resist a good math problem regardless of how depressed I felt about my current situation.

So I had a mall agency asking for a rental rate of

N percent of our gross sales each month. I had a fixed amount of utilities charges, and I not only had a question of how to perform the negotiations, but also a question of how the stall was supposed to operate under these conditions.

The first thing I did, therefore, was to take a look at one of the bakeshop's two other branches. For its part, this one was located in the middle of a supermarket, and was my closest comparison to a mall concessionaire that would conceivably pull in at least the same number of customers each day. Said supermarket stall averaged about

S pesos' worth of sales each month in 2008.

Now I needed to figure out how much profit we made from the supermarket branch; this would be our minimum target profit from the mall concessionaire. The way I figured it, I had the equation:

S

_{s} = M

_{s} + R

_{s} + U

_{s} + E

_{s} + P

Where:

S

_{s} = Gross sales of the supermarket branch per month;

M

_{s} = Material cost for the supermarket branch per month (i.e. raw materials);

R

_{s} = Rental for the supermarket branch per month;

U

_{s} = Utilities for the supermarket branch per month;

E

_{s} = Other expenses for the supermarket branch — such as transportation, payroll, spoilage, insurance, and zombie repellent — per month;

P = Profit (!) per month.

Given this, E

_{s} + P = S

_{s} - (M

_{s} + R

_{s} + U

_{s}). I had figures for everything but P and E

_{s}, but the value of (E

_{s} + P) at least gave me a base idea of how much money we wanted to make as profit from the mall concessionaire.

Now it was a question of following a similar equation for the mall concessionaire:

S

_{m} = M

_{m} + R

_{m} + U

_{m} + E

_{m} + P

Except that, since the rental value R

_{m} would be N percent of gross sales, I could easily do a replacement here:

S

_{m} = M

_{m} + (N/100)S

_{m} + U

_{m} + E

_{m} + P

I knew what our raw material cost M

_{m} was, what the mall's utilities charges U

_{m }would be like, and I knew what profit P we wanted, based on the supermarket branch. The only value I didn't have was E

_{m}, so I had to assume that we had an equal amount of expenses for both branches. Thus E

_{m} = E

_{s}, and therefore E

_{m} + P = E

_{s} + P, and I could get the value for E

_{s} + P from my earlier calculations.

As a result:

S

_{m} = M

_{m} + (N/100)S

_{m} + U

_{m} + E

_{m} + P

S

_{m} - (N/100)S

_{m} = M

_{m} + U

_{m} + E

_{m} + P

(1 - (N/100))S

_{m} = M

_{m} + U

_{m} + E

_{m} + P

S

_{m} = (M

_{m} + U

_{m} + E

_{m} + P)/(1 - (N/100))

For the mall concessionaire, given that I had an estimate for raw material costs (M

_{m}), its utilities charges (U

_{m}) and an idea of how much money I wanted to make (E

_{m} + P), I could easily calculate the needed mall sales S

_{m} from there.

As an example, let's suppose that we used up about Php 40,000 in raw materials each month for a given branch, that the mall would charge us 25% of all gross sales per month as rental, that the mall utility charges were Php 2,000.00 a month, and that the profit + additional expenses from the supermarket stall that we wanted to make was Php 50,000.00. That meant:

S

_{m} = (M

_{m} + U

_{m} + E

_{m} + P)/(1 - (N/100))

S

_{m} = 40000 + 2000 + 50000/(1-0.25)

S

_{m} = (92000)/(0.75)

S

_{m} = 122666.6667

...Which means that, under those circumstances, we would need to attain gross sales of about Php 123,000.00 a month to make about the same profit that we were getting from the supermarket.

That left the question of how far we could negotiate the percentage rate of rental. To find that, I just assumed that our gross sales for the two branches would be the same, and calculated the corresponding percentage based on that:

(1 - (N/100))S

_{m} = M

_{m} + U

_{m} + E

_{m} + P

1 - (N/100) = (M

_{m} + U

_{m} + E

_{m} + P)/S

_{m}1 - ((M

_{m} + U

_{m} + E

_{m} + P)/S

_{m}) = (N/100)

100 - 100((M

_{m} + U

_{m} + E

_{m} + P)/S

_{m}) = N

...Where S

_{m} = S

_{s}.

For another example, let's suppose that our supermarket gross sales were Php 100,000.00 a month. Let's also assume that our raw material cost held steady at Php 40,000 a month, that the mall utility charges were still Php 2,000.00 a month, and that the profit + additional expenses from the supermarket stall that we wanted to make was still Php 50,000.00.

100 - 100((M

_{m} + U

_{m} + E

_{m} + P)/S

_{m}) = N

100 - 100((40000 + 2000 + 50000)/100000) = N

100 - 100((92000)/100000) = N

100 - 100(0.92) = N

100 - 92 = N

N = 8

...Which means that any rental rate lower than 8% would have been better for us. Thus we would have started somewhere around this area — perhaps 5%, with the intent of maxing out at around 8% or so.

As it turned out, however, the rental rate was non-negotiable. The space was still attractive, however, and the additional sales we would need to make didn't look much higher, so we elected to plan it out further. As I write this, we're now in the middle of acquiring better expense data in order to improve my calculations; this should give us a solid sales quota per month that we can target.

Eventually we'll have to watch this as time goes on. Our initial sales are likely to fall short of target, for example, as people get accustomed to our presence. Then, once we get a small but steady group of buyers, we need to track whether or not we're gaining back our losses, or possibly check the rate at which we're gaining this per month. From there, it'll be a matter of sales charts, business forecasts, and other possible logical progressions.

Today, however, it's just about the theoretical algebra. And that's what I do, really.