Decompose a Recipe into Its Constituent Ingredients

[Rocky9242]

In the US, food products have an ingredient list and a "Nutrition Facts" panel.

The ingredient list lists all ingredients in order from most abundant to least abundant, e.g. "flour" would be the first ingredient for bread and "salt" would be further down the list.

The Nutrition Facts shows the abundance of macronutrients (protein, carbohydrate, fat) and micronutrients (minerals, vitamins) in the product.

Reliable nutritional information on ingredients is readily available online from the US Government.

What calculations would be necessary to reconstruct a recipe from nutritional data on the product and its ingredients and the ordered list of its ingredients?

So, maybe your bread contains flour, sugar, butter, yeast, and salt, listed in that order. You would start with the nutritional content of each of the ingredients and the nutritional information on the product and calculate the amount of ingredients in the bread, i.e. 6 cups flour, 2/3 cup sugar, 1/4 cup butter, 1-1/2 tsp salt. Assume the yeast has no effect on the figures and use the amount of yeast recommended on the package.

 

[dmytro]

Look at it this way: suppose you calculate the amounts of all the ingredients in the bread recipe. If you mix all the ingredients in a blender and put the thing into the oven, it's now easy to calculate the probability that you'll get a nice crispy bread, just use this formula:

$$P(\text{nice bread} | \text{no temporal info}) = 0$$

 

[Rocky9242]

Agreed. A person wishing to use the ingredients in these proportions would have to know how to cook them.

 

[Stephen Tashi]

A first try is to let $W$ be a matrix that gives a table of data for how many units of each nutrient are in each ingredient. Let [itex] W[j] [/itex] be the amount of nutrient $i$ that is in ingredient $j$. Let $x$ be a column vector whose entries are the unknown amounts of each ingredient. Let $s$ be a column vectors whose entries are the known totals of each nutrient. The system of linear equations to solve is $W x = s$.

Such systems of equation do not always have a unique solution. If we have ordered the ingredients accoring to their amounts, we can add the constraint $x[1] \ge x[2] \ge x[3] \ge ...$. We could make a more complicated mathematical description of the problem by considering further practical realities.

 

[blue_raver22]

To reconstruct the recipe, you would need to first determine the total weight of the product. This can be done by multiplying the serving size by the number of servings listed on the Nutrition Facts panel. Next, you would need to calculate the percentage of each macronutrient and micronutrient in the product. This can be done by dividing the amount of each nutrient by the total weight of the product and multiplying by 100.

Once you have the percentage of each nutrient, you can use the ingredient list to determine the amount of each ingredient in the product. For example, if the bread contains 6 cups of flour and the flour accounts for 50% of the total weight of the product, then the total weight of the product is 12 cups. Using this information and the percentage of each nutrient in the product, you can calculate the amount of each nutrient that comes from the flour. This process would need to be repeated for each ingredient in the recipe.

It's important to note that this method may not be entirely accurate, as some ingredients may contribute to more than one nutrient. For example, butter contains both fat and protein, so the calculations for these nutrients would need to take that into account.

Additionally, this method assumes that the ingredients listed on the product label are the only ingredients in the recipe. If there are any additional ingredients or variations in the amounts used, this could affect the accuracy of the reconstructed recipe.

Overall, while it is possible to use nutritional data and ingredient lists to reconstruct a recipe, it may not always be completely accurate and may require some adjustments and estimations.