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Ingredient Math: Calculating minimum ingredient proportions

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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.

Talenti Chocolate Coffee Chocolate Chip – company response on caffeine content and natural flavors query

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I had previously reviewed Talenti’s Chocolate Coffee Chocolate Chip ice cream, which maintains first place for my most loved ice cream. In that post I discussed sending an email to the company requesting more information about this product, and since I received a response from them I decided to write it up as a new post.  The original blog post is here for those interested.

I had requested two things from them: caffeine amount and detailed explanation of “natural flavors”. The former was because I had felt quite a ‘kick’ from eating this and wanted to determine whether that was from sugar, caffeine, or something else. I asked the latter from my uneasiness as to what I am actually eating. It’s apparently ‘natural’ but what is it really? Consumers who want to research more about the possible side effects and nutrition of this catch-all ingredient are at a loss.

First I’ll give an excerpt of the polite email I received from Talenti, followed by my comments on it.

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Thank you for the inquiry.  We do understand your concerns regarding “natural flavors’ in our ingredients.

Unfortunately, the natural flavor(s) in our ingredients are considered a trade secret and is proprietary information. I can tell you the consist of juice, juice concentrates, essences, essential oils, and extractives.
 
Also, the amount of caffeine in our product is less that 2 one hundredths of one percent.  Our Coffee Chocolate Chip gelato has 5 – 8% per serving.
 
I hope this helps and apologize for not being at liberty to share more information with you.
 
Please feel free to contact us if you have any further questions or concerns. We wish you all the best,
 
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According to the response, caffeine is only 5-8% per serving, which means only 5 to 8 milligrams per 102 gram serving. I typically eat half a pint (two servings) which translates to at most 16 mg of caffeine, far from enough to feel anything for anyone with a moderate caffeine tolerance. This leaves the cause of the buzz I was feeling to come from sugar or something else in the product.
Because of competitive reasons Talenti gave me practically no information on what is contained within “Natural Flavors”, only that it consists of “juice, juice concentrates” (which I am generally OK with), as well as “essences, essential oils, and extractives”. This latter group is still quite vague and could be practically anything. The only consolation we have is that since this ingredient is listed 10th on the label, we know there is no more than 10% of it present in the product (see here for a post on where I got this number from).
I don’t think it will do much good to push Talenti for more information at this stage, but I haven’t given up on exposing what is really in our foods.
Part of me wonders how much more can be discovered by doing a lab analysis of the product, but surely there is a high cost there so I’ll put that off until another time.
I really enjoy learning more about sweet treats directly from the producer, and hope to continue to provide this type of information in this blog.