# More on Proportional Reasoning

A proportion is a part, share, or number compared to the whole, for example:

• The proportion of greenhouse gases in the atmosphere is rising.
• Only a small proportion of the land can be farmed.

Note that the comparison is multiplicative, for example:

• The proportion of sugar to water in soft drink is $1$ part sugar to $9$ parts water. There is nine times more water than sugar.

We usually express proportions as fractions, ratios and rates.

#### Ratio

Here are some examples of proportions in ratio form.

• a part to the whole
$1$ out of every $3$ eggs are white.
We write '$1:3$'.
We say '$1$ is to $3$'.
Notice that one third of the total number of eggs are white.
• part to part (of the whole)
For every brown egg there are $4$ white eggs.
'$1:4$' or '$1$ is to $4$'.
Notice that one fifth of the total number of eggs are brown.

Remember proportions descrive a multiplicative relationship between quantities. For example:

Orange paint is made by mixing red and yellow in the ratio $1:3$. If I added two more parts of red, how many more parts of yellow would I need?

I would add $6$ parts so that the multiplicative relationship is preserved (that is I need three times as many parts of yellow as I need parts of red in the mix).

#### Rates

A rate is a ratio that compares different types of quantities.

Here are two examples of rates:

• You walk $5$ km (distance) for each $1$ hour (time). We say '$5$ km per hour'. We write '$5$ km/h'.
• There are $70$ people for each square km (population density). That is $70$ people/square km.