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Ingredient Math: Calculating minimum ingredient proportions

In a previous article here, I showed how to calculate the maximum proportion of a ingredient listed on a food label. I derived the following simple formula:

Maximum percentage of the Nth ingredient = (100 / N)

This time I’ll use similar logic to derive a formula for the  minimum amount of an ingredient, based on only its order in the listing and the total number of ingredients.

Let’s start with the simple case of a product with two ingredients: “sugar, cocoa”.

With a little thought, we can see that sugar cannot be any less than 50%. If it was, then cocoa would have to be greater than 50% (since it equals 100% – sugar) and that is impossible based on the ordering rule that the most prominent ingredients, by weight, are listed first.

What about cocoa? Clearly it has no minimum since it could be 0.00001%, leaving the remaining 99.9999% to sugar.

Next let’s try a product with three ingredients: “sugar, cocoa, shredded coconut”.

Using similar logic, we will see the minimum amount of sugar is 33%, since any less than that would mean the other ingredients would have to have a higher proportion than sugar. If we assumed sugar was 20%, then the other two ingredients must add up to 80%, and therefore one of them must be present in at least 40% proportion – but that is greater than sugar which is listed first!

And the second ingredient? Just like the second ingredient in the two-ingredient example, this can be arbitrarily small. For example: 99.9% sugar, 0.09% cocoa, and 0.01% shredded coconut.

To generalize these results:

Minimum percentage of the 1st ingredient = (100 / T), where T is the number of total ingredients

Minimum percentage of the other ingredients = infinitesimally close to zero

As with the calculations of maximum proportion in my previous article, if we have additional information about the other ingredients we can use that to determine the minimum level of the other ingredients. For the same example, if we sifted and then weighed the shredded coconut (third ingredient), we could determine its overall proportion by weight. Lets say this is 10%. Then we could reason that cocoa, the second ingredient, takes up a minimum of 10% total weight since it must be equal or greater to the amount of shredded coconut. From this we could also conclude sugar (the first ingredient) is at most 80% of the total weight.

You can use nutritional values such as protein and fiber to infer how much of certain ingredients are present, especially if those have a large proportion of a certain element.

You may feel that this sort of calculation may seem unlikely to have any practical use, but if you find yourself saying “I eat so-and-so product because it contains a large amount of X, which is healthy”, then you can use it to see the minimum level of that ingredient you are consuming.

Another use is if you are trying to make a homemade version of something and you want to get a feel for the minimum and maximum amount of it in the product.

Ingredient Math – estimating worst (or best) case for ingredient proportions

Have you ever wondered how much of a certain ingredient is really present in a sweet, or any food product?

You probably know that ingredients are listed on food labels in order or prevalence, with the most predominant ingredient first. You may have even known this was determined by weight. But in this article I will discuss a method to get an estimate for the maximum of each ingredient’s percentage of total weight – just by using the ordered ingredient list.

To derive this formula, lets start with a very simple example, a product with just “coffee and sugar”. Since coffee is listed first we know it has higher or equal amount of total weight when compared to sugar.

Is there anything we can do to determine about how much the first ingredient, coffee, is really in the product? The answer is no because coffee could be almost 100% to almost 0% of the total weight, with sugar filling in the remaining space. (Actually, there is a trick to determine the amount here since the second ingredient is sugar, which I’ll discuss later in this article).

Well, if you think about it, there can’t be more than 50% sugar, by weight, since any more of that would mean there was more sugar than coffee, which we know is not the case.

So we’ve learned something important – that there is no more than 50% sugar in the product. This would apply to another second ingredient when there are two total ingredients.

What if there were three or more total ingredients? We would get the same result, because the other ingredients could be in trace amounts (practically 0%), so the “50% maximum for the second ingredient” rule would still apply.

What about the maximum amount of the third ingredient? Using the same logic you will see it cannot be above 33.3%, since any more of that would mean it is in greater proportion than the first and second ingredients. And for the forth ingredient you get a maximum, by weight, of 25%.

Turning this into a simple formula we get the following:

Maximum percentage of the Nth ingredient = (100 / N)

So for the 5th ingredient, you would get (100 / 5) = 20% maximum weight of that ingredient.

If you use formula along with the serving size you can determine the maximum weight of any of the ingredients per serving. Pretty handy if you want to minimize your intake of certain things.

If you want to take this to the next step, you can infer more information when one more more ingredients are a type of sugar. For example, if a product contains “coffee, sugar” and has 3 grams of sugar per 15 gram serving, then you know right away there is 20% sugar and 80% coffee in this product. Keep in mind that the grams of sugar listed includes any type of sugar, so if you have multiple ingredients which contain some type of sugar (even fruits) then the calculation gets a little trickier.

Besides knowing there is a certain percentage of sugar, you can use that to deduce information about other ingredients.

For example, if the imaginary product I just described had a third ingredient, say “coffee, sugar, vanilla”, then you would know that there is 20% or less vanilla because sugar is 20% or less. This assumes that there is no sugar in the vanilla, otherwise it would be harder to make any definitive conclusions.

Similarly, if you know how much protein is in each ingredient, you can figure out even more using the supplied protein in grams.

You can also leverage information about other ingredients to deduce additional information about the other ingredients. For example if a product had “milk, sugar, guar gum, vanilla”, you would know that the proportion of vanilla is much less than 25% since guar gum is typically used in relatively small doses. (I’ve tried overusing guar gum in homemade ice cream – its not pretty!)

I love thinking about food and ingredients from a methodical, logical point of view since it allows me to apply science to my everyday life.

References

http://www.fda.gov/Food/GuidanceRegulation/GuidanceDocumentsRegulatoryInformation/LabelingNutrition/ucm064880.htm