Cross-multiplication

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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:
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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:
$\frac{7 \times11}{2} = x = 38,5L .$

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.

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A delicious example of cross-multiplication

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