# Decompose a Recipe into Its Constituent Ingredients

1. Aug 12, 2014

### 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.

2. Aug 12, 2014

### 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:

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

3. Aug 13, 2014

### Rocky9242

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

4. Aug 14, 2014

### 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 $W[j]$ 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.