Scaling a recipe — the factor from servings or from an ingredient

The left column holds how many people the recipe serves (top) and how many you are cooking for (bottom). Factor, between them, divides the second by the first. The right column is the recipe as written: four of the ingredients from a recipe for meatballs that serves 4. The middle column is what you actually use, each worked out as "recipe amount × factor".

Changing the number you are cooking for updates the factor and all four ingredients. With a different recipe's amounts in the right column, the same board scales that recipe.

The factor is servings wanted ÷ servings in the recipe

Cooking for 6 instead of 4 does not change how much each person gets. Only the number of people changes, by 6 ÷ 4 = 1.5 times. With the same amount per person, every ingredient grows by the same 1.5 times.

amount to use = recipe amount × (servings wanted ÷ servings in the recipe)

Working out one serving first and then multiplying gives the same answer. For the meat, one serving is 400 ÷ 4 = 100 g, and six servings are 100 × 6 = 600 g. Working out the factor first means one division in total; after that each ingredient is a single multiplication by the same number.

On Board 1 there is only one factor card, and all four ingredient cards use it. Changing the number of people changes the factor, and the new factor applies to all four ingredients at once.

The factor depends on how many servings the recipe is written for. With Recipe serves set to 2 on Board 1 (still cooking for 6), the factor becomes 3, the meat 1,200 g and the salt 12 g. Treating a recipe for 4 as a recipe for 2 doubles every ingredient compared with the right amounts for 6.

Taking the factor from one ingredient

Sometimes the fixed quantity is an ingredient, not the number of people, such as a whole pack of meat to be used up. Then the factor comes from that ingredient.

factor = amount you have ÷ amount in the recipe

With 500 g of meat, the factor is 500 ÷ 400 = 1.25, and every other ingredient is multiplied by the same 1.25.

Factor, in the centre, divides Meat you have (right) by the recipe's meat (left). Now serves, at the top, multiplies the recipe's 4 servings by the factor: 5 servings. The two lower rows are the onion and the eggs: the bottom row is the recipe amount, the row above it the scaled amount. The onion comes to 250 g and the eggs to 1.25.

With Meat you have set to 600 g, the factor is 1.5, for 6 servings, with 300 g of onion and 1.5 eggs: the same amounts as Board 1 cooking for 6.

The factor is the same whether it comes from people or from an ingredient, because every amount in the recipe is proportional to the number of servings. If the meat is 1.5 times the recipe, so is the number of people it feeds.

Why one factor keeps the recipe in balance

The reason for multiplying every ingredient by the same factor is to keep their proportions. How salty something tastes, for example, is not set by the amount of salt alone, but by how much salt there is relative to everything else.

The top row is the recipe: 400 g of meat and 4 g of salt. Salt ÷ meat, in the middle, is the salt as a percentage of the meat: 1.00%. The middle row multiplies both by the factor of 1.5, giving 600 g of meat and 6 g of salt. The card at the bottom is the same percentage after scaling, and it is 1.00% too.

Multiplying the top and the bottom of a division by the same number does not change the result. In (4 × 1.5) ÷ (400 × 1.5) the two 1.5s cancel, leaving 4 ÷ 400. With the factor on Board 3 set to 0.5, 2 or 3, the percentage stays at 1.00%.

Increasing only the meat and leaving the salt as it was does change the proportion. With the meat in the top row of Board 3 set to 600, Salt ÷ meat drops to 0.67%, about two thirds of what it was.

The same goes for the onion and the eggs. For any two ingredients multiplied by the same factor, the ratio between them stays the same.

Other numbers

Board 1, changing the number of people you cook for (the recipe serves 4):

Each extra person adds 0.25 to the factor, because in a recipe for 4 one serving is a factor of 0.25. In this table the eggs only come out as whole numbers when cooking for 4 or 8.

Board 2, changing the meat you have (the recipe uses 400 g and serves 4):

Every extra 100 g of meat adds 0.25 to the factor and one more serving, because this recipe uses 100 g of meat per person.

What the factor does not decide

The factor sets the amounts of the ingredients. Some things in cooking do not grow in proportion to the amounts.

Ingredients you count — 1.5 or 1.25 eggs cannot be used as they are. Beaten eggs can be measured out in part. Rounding to one or two whole eggs instead gives the eggs alone a different factor, so their ratio to the other ingredients shifts a little.

Seasoning and liquid — the 6 g of salt on the board is the amount that keeps the same proportion to the meat. In dishes with liquid, such as stews and sauces, water evaporates while they cook, and how much evaporates depends on the width of the pan, the heat and the cooking time. So scaling every amount in proportion does not guarantee the finished dish tastes as strong as the original. How much to adjust varies from dish to dish, and this calculation does not decide it.

Cooking time and equipment — how long something takes to fry or simmer is not set by the total amount alone. The size and thickness of each piece, how full the pan is and the heat all play a part, so multiplying the time by the factor does not necessarily give the right time. Nor will meatballs for 8 necessarily fit in the pan in one go.

Read next — Concentration — finding it, diluting and mixing It calculates a concentration by dividing the weight of salt by the weight of the whole solution, and also covers diluting with water and mixing two solutions.

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