Math Relationships Cross-multiplication

# Cross-multiplication

The cross-multiplication is a solution method for tasks with (inversely-) proportional sizes.

For cross-multiplication exercises, one looks for a certain size with given values. You then try to increase or decrease the values to the size you are looking for.

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### Method

1. Is it a proportional or inversely proportional relationship?
2. Calculate auxiliary value (usually 1)
3. Convert auxiliary value to searched size

## Proportional relationship

For proportional relationships applies:

„the more the more

and vice versa

„the less the less

We imagine "more" as multiplication and "less" as division. The same arithmetic operation must be carried out on both sides.

### Example

$\begin{array}{c|c|c|c} \text{Number} & & \text{Price} & \\ \hline 10 & \color{red}{:10} & 30 & \color{red}{:10} \\ 1 & {\color{green}{\cdot4}} & 3 & { \color{green}{\cdot4}} \\ 4 & & 12 & \end{array}$

A little more detailed explanation:

1. #### Proportional or inversely proportional relationship?

It's a proportional relationship, because the more breads you buy, the more you have to pay.

2. #### Enter values and determine the auxiliary value

We enter the given sizes.

$\begin{array}{c|c} \text{Number} & \text{Price} \\ \hline 10 & 30 \\ 4 & ? \end{array}$

We calculate the price per piece, i.e. we calculate the value to 1:

$\frac{\text{Price}}{\text{Number}}$ $=\frac{30}{10}$ $=\color{blue}{3}$

3. #### Calculate the size you are looking for

The unit price will now be multiplied by the number.

$\text{Number}\cdot$ $\text{Unit price}$ $=4\cdot\color{blue}{3}$ $=12$

## Inversely proportional relationship

For inversely proportional relationships applies:

„the more the less

and vice versa

„the less the more

We imagine "more" as multiplication and "less" as division. On both sides, therefore, the reverse arithmetic operation must be performed.

### Example

3 workers need 15 hours. How many hours do 5 workers need?

$\begin{array}{c|c|c|c} \text{Workers} & & \text{Hours} & \\ \hline 3 & \color{red}{:3} & 15 & \color{green}{\cdot3} \\ 1 & {\color{green}{\cdot5}} & 45 & { \color{red}{:5}} \\ 5 & & 9 & \end{array}$