Mathematics for start-up brands

Most weeks I talk a client or two through some basic mathematics so that they can scale-up and cost their formulations.  So many people seem to struggle with this aspect of their crafting/ formulating/ business planning that I reckoned it was worth a blog post so here it is.

Note: I am NOT the words most gifted mathematician,  in fact I pretty much sucked at all the math part of my Degree and I’ve hardly touched the subject since but I do know percentages and that’s where we are going to start.

So when you first make your formula (or recipe if you are a crafter) you might use a range of different measurements.  This is a big science no, no but hey, let’s not be too judgy about that.

OK, your recipe might look a little like this for example:

1 cup of  coconut oil

1 cup of water.

10 drops of lemon essential oil

5g emulsifying wax

1 tsp honey

Now aside from the fact that the above would be a pretty crap recipe we also can’t get a handle on just how crappy it is unless we can ‘see’ it properly. The way we see things in cosmetic chemistry is in percentages.  Let’s do this thing.

First we have to unscramble our mixed units.  We have cups, drops, spoons and grams.

So cup measurements do vary so it would be useful to investigate your ‘cups’ before settling on a volume but we’ll stick with the metric as that’s what I grew up with.

1 cup = 250ml

Drops are even more ambiguous than cups as the weight of a drop depends on the oils specific gravity – this term will come up more than once.

Specific Gravity is a ratio of how the ingredient relates to water. Water has an SG of 1.  1 = equivalence.  1g of water takes up 1ml of space. Most liquids are different to that with some liquids being heavier than water (salt water, glycerin, sugar solution etc) and others being lighter (pretty much all the oils).   Vegetable and essential oils all have specific gravities of less than 1 so 1 litre of oil is always less than 1Kg.

Lemon oil has a specific gravity of between 0.849 – 0.855

Sandalwood Oil has a specific gravity of 0.965-0.980

Coconut oil has a specific gravity of around 0.927

Sweet Almond Oil has a specific gravity of 0.91-0.92

Castor Oil has a specific gravity of 0.959

Specific gravity values are usually presented as a range on a product specification as they can subtly change from batch to batch.  Further, specific gravity is temperature dependent with the range values usually relating to readings at room temperature (typically 25C but can be 21C if measured in cold climates).

The ramifications of that are that in this recipe even though there is one cup of water and one cup of oil the weights of the cup contents will differ.

Now it’s time to start writing the recipe in grams so that we can scale-up (no factory measures large batches in volume, it’s always weight).

1 cup of water = 250 g

1 cup of coconut oil is a bit trickier.

So we have 250ml of coconut oil (1 cup).

250 (coconut oil in mls) x 0.927 (average specific gravity of the oil) = 231.75.  So we have 231.75g of coconut oil in this recipe.

10 drops of essential oil.  Drops are quite tricky to convert so we have two choices:

1. get some mini scales and actually weigh out what a drop (or 10 drops) weighs then just use that number in your formula.
2. Refer to Tisserand and Young’s book ‘Essential Oil Safety’ which does give a conversion table in Appendix C.  In there it states that 9 drops is around 0.3ml and 12 drops is around 0.4ml so we could guess that 10 drops is around 0.33ml. We then just have to convert that into g using the specific gravity calculation:  0.33 x 0.852 (the average specific gravity) = 0.28g

The emulsifying wax is already in grams thankfully (5g) but the honey isn’t.

The honey is 1 tsp.  A teaspoon in metric is 5ml.

Honey has a specific gravity of 1.035 which means it is HEAVIER than water.  So 5ml is going to work out to be 5 x 1.035 = 5.175g

So now our recipe looks like this:

231.75g coconut oil

250g cup of water.

0.28g drops of lemon essential oil

5g emulsifying wax

5.175g honey

 

We now add these up to get the very weird total of 492.205g.

So our 100% recipe = 492.205g

We want 100% of the recipe to equal 100g and to make that happen we have to do a bit more math.

It is best if 1% = 1g for the recipe as that’s the easiest and quickest way to scan a recipe for error. You can make whatever batch size you like but we must start from a simple place.  I do that this way:

 

100 (what we want the recipe to add up to) / 492.205 (what the recipe adds up to now) X each ingredient in turn = …

So:

For coconut oil 100/492.205 * 231.75 = 47.09 g OR % – now they are the same, 1g=1%.

Water: 100/492.205*250 = 50.8g OR %

Lemon: 0.06g or %

Emulsifying Wax 1g or %

Honey: 1.05g or %.

TOTAL 100 g / %

Let’s write that again without all the background noise:

Coconut Oil 47.09%

Water 50.8%

Lemon 0.06%

Emulsifying wax 1%

Honey 1.05%

TOTAL 100g

 

Once the recipe is in this format it is easy to see what’s what. I just made this recipe up and it would not work because there is not enough emulsifier (I suspect although I haven’t specified what the emulsifier is in my fictitious cream).  Also there is no preservative AND the oil:water ratio is very close and that can often cause a bit of an issue.

But this is about math not formulating.

If you did get to this point and decided you wanted to add a preservative (as you absolutely should) you might be thinking ‘well how can I do that now, I have 100%, I can’t add preservative and then have a recipe totalling 101% now can I?

What you would do is this.

Your preservative selection information should outline how much preservative is recommended.  Let’s say it’s 4% so you know you have to end up with 4% preservative in your formula.

You have a couple of options.

Either you can reduce the water or oil phase by 4% and then use that gap to place the preservative. That is as simple as just changing the figure of the water or coconut oil -4 then adding the preservative to the bottom of the recipe. I’ll adjust the water:

Coconut oil 47.09%

Water 46.8%

Lemon 0.06%

Emulsifying wax 1%

Honey 1.05%

Preservative 4%

TOTAL 100% 

The downside to that method is that now you have changed the ratio of oil phase to water phase. This may change the overall stability of the formula and/or the look and/or feel.  Maybe this isn’t such a good idea.

Another way to do it is to scale back the whole formula.  This is more math but still worth it as it maintains the ratio relationships while still making room for the preservative.

To do that we just pretend the mixture now adds up to 96% rather than 100% (or 96g instead of 100g.  That is  done using the following calculation:

Ingredient current % / 100 * 96 =

Coconut Oil 47.09 / 100 * 96 = 45.21

Water 50.8% = 48.77

Lemon 0.06% = 0.058

Emulsifying wax 1% = 0.96

Honey 1.05% = 1.008

TOTAL 100g = 96g or %

Now there is room to pop the 4% of Preservative in making the formula.

Coconut Oil 45.21%

Water 48.77%

Lemon Oil 0.058%

Emulsifying wax 0.96%

Honey 1.008%

Preservative 4%

TOTAL 100%

If you are buying base products from a supplier and adding your stuff to them you are, in effect, diluting down the base evenly with your new inputs and making room for your new ingredient.

Rather than scaling back the formula mathematically you could do it the other way and just add more on, the net result may be a little different but that may not matter too much, it depends on how much stuff you are adding.

Say we started with our original formula in and just added 4% of preservative to it to give us a total of 104%

Coconut Oil 47.09g

Water 50.8%

Lemon 0.06%

Emulsifying wax 1%

Honey 1.05%

Preservative 4%

TOTAL 104%

Only doing it this way we don’t end up adding 4% of preservative, we end up with our additional preservative being LESS than 4% as a recalculation of our formula back to 100% shows us.

Coconut oil 47.09 / 104 * 100 = 45.28%

Water 50.8 / 104* 100 = 48.85%

Lemon 0.06 / 104*100 = 0.058%

Emulsifying wax 1/104*100 = 0.96%

Honey 1.05/104*100 = 1.01%

Preservative 4/104*100 = 3.85%

So what we thought was a 4% addition was only a 3.85% addition, not a dramatic difference here but still, it’s not the same.  The reason for this is that when you add something to the recipe everything gets adjusted as a recipe ALWAYS has to be back-calculated to 100%.

You can see in this example that the calculation totals from making the base 96% to pro-actively make room for the preservative at 4% resulted in slightly different ratios for some ingredients vs the example where we reactively back-calculated the totals.  It’s important to be aware of that and also to the fact that the more you add to a formula, the larger the impact will be.

I hope that makes some sense for you but if it doesn’t don’t worry, it might make more sense when you look at your own recipes or if you even go through the motions in your own lab.  The bottom line though is that you can’t accurately cost a formula that isn’t in a single set of units and it is easier to trouble shoot, cost and scale-up a formula when the ingredients all add up to 100%.