Mathematics and cooking

Summary: A good cook must be able to compute conversions, costs, and measurements.

In his Renaissance play, The Staple of News, Ben Jonson likens a master cook to—among other things—a mathematician. Although many people would think this comparison is an exaggeration, the mathematical requirements placed on the modern cook are significant.

In the past, cooking skills were passed on orally and through apprenticeship from generation to generation; today, inexperienced cooks are expected to learn to cook from recipes, which consist of a list of measured ingredients followed by instructions that refer to temperatures, times, and possibly more esoteric measurements. In addition to being able to scale recipes, the cook in our global world encounters many interesting recipes from diverse cultural traditions, which use a variety of systems of measurement. Cooks must also be able to plan healthy and cost-effective menus.

Measurement of Ingredients

In recipes written in the United States, quantities for both liquid and dry ingredients are often specified by units of volume, and are measured in terms of teaspoons, tablespoons, or cups, in which there are 3 teaspoons to 1 tablespoon, 16 tablespoons to 1 cup, and 2 cups to 1 pint. Special measuring cups are made that permit the leveling of dry ingredients to ensure precise measurement. For measuring liquid ingredients, different cups are used that have graduation marks down the side and a convenient pouring spout. Measuring spoons are used for smaller quantities of both liquid and dry ingredients. For an experienced cook, the quantities given in recipes serve as general indications; however, in baking, when certain chemical reactions are expected to be balanced, precision is needed.

For more consistent outcomes, quantities are specified by weight. Ingredient densities vary. For example, a cup of water weighs 8 ounces, whereas a cup of flour—depending on how it was scooped—weighs about 5 ounces. Tables to assist in conversion between weight and volume can be found on the Internet. There can be confusion with the word “ounces,” which can refer to either weight or volume. Ounces used for dry ingredients refer to one-sixteenth of a pound. Ounces measuring liquid ingredients refer to either one-sixteenth of a pint or to one-twentieth of a pint, depending on what is being measured.

Modern recipes written outside the United States provide measurements in the metric system. Liquid ingredients are specified in liters (volume) while dry ingredients are specified in grams (mass). Since kitchen scales actually measure weight, most cooks view grams as measuring weight. One liter of water weighs approximately 1000 grams. A liter is 1000 cubic centimeters, or about 1.057 quarts. A kilogram, 1000 grams, is approximately 2.205 pounds. A deciliter is one-tenth of a liter and is often used for recipes designed for home use. The metric system—based on multiples of 10—is designed to simplify calculations and scaling of measurements and is becoming the preferred system for cooks.

Scaling a Recipe

Recipes often specify the number of portions that they produce. To alter the number of portions generated, the recipe is scaled. This involves multiplying the quantity of each ingredient by a scale factor. To double a recipe, the scale factor is 2, while to halve a recipe, the scale factor is 1/2. At times, a more complex scaling is required. For example, imagine a baker is following a recipe that calls for 125 grams of pre-fermented dough. The recipe to make pre-fermented dough calls for 1000 grams of flour, 10 grams of yeast, and 0.6 liters of water and results in 1610 grams of dough. Since only 125 grams of pre-fermented dough are needed, the required scale factor is 125/1610 = 0.078.

A naïve scaling results in 78 grams of flour, 47 grams of water (.047 liters), and the absurdly small amount (0.78 grams) of yeast. An experienced cook would add more yeast. Most recipes written for home use can only be scaled up or down by less than a factor of 4. Additionally, some ingredients, like spices, gelatin, and leavening agents, should not be scaled proportionately. Most good general cookbooks will give advice on scaling recipes. A good collection of professional recipes for large numbers of portions is available from the Armed Forces Recipe Service.

Measuring Temperature

Controlling temperatures on most modern stovetops is easier than doing so on wood-burning stoves. However, techniques vary significantly among gas, electric, and induction cookers and are best described by the manufacturer. In some instances, such as deep fat frying or candy making, temperature on the stovetop is measured by a thermometer. In making candy sugar syrup, temperature can also be measured by “feel” or by the way a drop of the syrup interacts with cold water. Books on making candy describe the relationships among these methods. The temperature of an oven is accurately monitored by a thermostat, which can be set. Often, an oven thermometer is also used to check the oven thermostat. Most recipes give the required temperature in either Fahrenheit or Celsius (previously called centigrade). The formula for converting from Fahrenheit to Celsius is given by

and from Celsius to Fahrenheit by

Thus, an oven temperature of 350 degrees Fahrenheit is about 177 degrees Celsius. Temperatures in some older British recipes are given in gas mark settings, in some older French recipes in numbered settings, in some older German recipes as Stufe settings, and in some much older recipes as verbal descriptions such as Very Slow or Doux. Tables showing conversions among these various approaches to measuring temperature can be found in general cookbooks and on the Internet.

Other Important Measurements

Other important quantities that need to be measured when cooking include time, acidity, and density. Time measured in seconds, minutes, and hours—a system based on 60—is now probably universal. Because estimating the passage of time is fraught with error, early recipes specified important times “as measured by the clock.” Acidity is measured on the pH scale. Water, which is neutral, has a pH of 7. An acidic solution, like orange juice, might have a pH of 3, while a basic solution of baking soda in water might have a pH of 9. In home cheese making, the conversion of lactose to lactic acid is tracked by monitoring pH levels of the milk; however, traditional cheese makers will use the Dornic scale.

Measuring the density of a solution is important in wine and beer making, and in candying fruits. For example, the density of fresh grape juice indicates the ripeness of the grapes and the alcohol content of the finished wine. Candying fruit in sugar water can take many days. The daily gradual increase of sugar in the syrup where the fruit is steeping maximizes the amount of sugar absorbed by the fruit. The density of the syrup is carefully checked to ensure the correct increase of sweetness. Density of syrups is measured with a hydrometer, and a variety of scales, including Brix, Baumé, and specific gravity, have been used in recipes. Although older French recipes will refer to the Baumé scale, since the 1960s, most recipes have used specific gravity. For syrups that are denser than water, a simple approximate conversion from Baumé to specific gravity (sg) is given by:

Menu Planning and Budgeting

Cost and nutrition are also important factors for cooks. Many modern recipes, in addition to giving calories per serving, will give grams of carbohydrates, protein, fat, cholesterol, sodium, and calcium. This information, along with labels on prepared food, helps guide the cook in making nutritional choices. A cook might also be interested in knowing the cost of a portion size. For example, consider a portion of boneless chicken breast. The cost as purchased is what the chicken breast with bone costs per pound. Once the breast has been boned, what remains weighs less and results in a higher cost per pound of the edible portion. During cooking, the breast will shrink, resulting in an even higher cost per pound of the breast as served. Being aware of these costs, along with labor costs and inventory costs, helps the cook determine the cost of each item served. Although the home cook probably does not go through all these computations, a good home cook will have an idea of monthly food expenditures and how these costs are distributed among the various kinds of food served.

Bibliography

Bilheux, R., and Alain Escoffier. Creams, Confections, and Finished Desserts. Hoboken, NJ: Wiley, 1998.

Haines, R. G. Math Principles for Food Service Occupations. 3rd ed. Albany, NY: Delmar Publishers, 1996.

Jones, T. Culinary Calculations: Simplified Math for Culinary Professionals. Hoboken, NJ: Wiley, 2004.

Labensky, S. R. Applied Math for Food Service. Upper Saddle River, NJ: Prentice-Hall, 1998.

Reinhart, P. The Bread Baker’s Apprentice: Mastering the Art of Extraordinary Bread. Berkeley, CA: Ten Speed Press, 2001.