You see, the liquor bottle recipes don't know if you are making a drink for a single person, or a crowd, or whether you want only a small-sized drink. So they give the proportions of drinks, and it's up to the consumer to decide what is the appropriate quantity.
So, a "part" is kind of like the x in your algebra book. First, off, when you are deciphering a recipe with "part"s, you need to see how many parts are included in the recipe. 5 parts? 10 parts? 35 parts?
Now envision the container for your final product: a double old-fashioned glass, a large beer stein, a punchbowl? Think about how much it takes to create the desired amount of your final product.
And now get out your measuring spoons or cups, and decide how many of those will fit into your container. When you get a close match between the size of your measuring equipment times the number of parts and your container size, that will tell you how much is a "part."
To make the equation work, then, you have a "part" which can be a teaspoon, a tablespoon, a cup, a pint, a gallon, or even more, or anything in between. Then you are going to use that measurement for everything that is a "part," whether you want one part, three parts, or twenty parts, or something else. It's all about how much of one ingredient in relation to how much of the other ingredient.
So if you have a recipe that calls for three parts soda to one part of rye, mentally divide your container (glass, pitcher, punchbowl) into four parts. If it's an eight-ounce glass you want to fill, you have four parts, so you'd divide that eight ounces by four, and you get two ounces (actually a little less as you'll need some room at the top). Then you take three two-ounce measures of soda, and one two-ounce measure of rye.
Why are proportions important in food? In many recipes, proportions are all you get, as we have seen in the liquor store examples. However, in any case, knowing the general proportions of recipes helps free you from the recipe books and from dragging out the measuring tools each time you want to make something, and that makes getting to the point where you're ready to dive into your food that much closer!