Concept history

Concept created

##### Intuitive definition
The cross-product is a simple calculation used to find a missing value based on three other values, when those values are related by a proportional relationship.
This calculation allows you, for example, to find the new value of a proportion when the total changes, or to adjust two values proportionally at the same time.

##### How to compute with an example
We refer to this as a “cross-multiplication” because of the traditional way of visualizing the calculation in a two-way table.
For example: We need $11L$ of substrate to fill two jardinieres and we want to know how many liters of substrate are needed to fill $7$ of them. We write the following table, where each column corresponds to a scenario based on the number of jardinieres:
![produit en croix eng](https://s3.pub2.infomaniak.cloud/object/v1/AUTH_7cc517879b0040959f7d12abb1f0e72d/mathwoods-images/uploads/2026/08/user-21/1787328586613-0973ca48638ed95d-produit-en-croix-eng.png#mw-width-88)
We always **multiply** the numbers arranged **diagonally** in the table (following the blue line), and we **divide** the product by the **last value** available (following the yellow line).
So we have:
$$ \frac{7 \times11}{2} = x = 38,5L  .$$

:::fold Caution!
The table can be arranged differently (by swapping the two rows, the two columns, or placing what is in a column into a row), but we must always be careful to clearly identify where the two situations with the same proportional relationship occur. It is often a matter of clearly visualizing the situation in which the calculation is being performed to understand which values are related to one another.
In our example, we therefore have:
$$\frac{2}{11} = \frac{7}{x} $$
but  
$$\frac{2}{x} \neq \frac{7}{11}.$$
:::

##### Examples
- We use a cross-multiplication to compute new quantities of ingredients in a recipe when the number of portions is not the same than in the original recipe.
- The conversion from a percentage to an absolute value (and vice versa) can be done using a cross-multiplication.