# Blog Archives

## 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.