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Fractions as ratios

Ratios are most commonly used to express the quantitative relationship between two groups, and can be modelled using discrete items.

For example, the ratio of 2 to 3 can be represented as a group of two squares compared to a group of three triangles:

Two squares of equal size grouped together on the left. Three triangles of equal size grouped together on the right.

Ratio of 2 to 3.

This can be expressed as 2:3 or as \(\frac{2}{3}\).

Another way of describing a ratio is that it indicates the number of times one amount contains the other amount, or is contained within the other amount. We could say that:

  • the group of squares is \(\frac{2}{3}\) the size of the group of triangles, or
  • the group of triangles is \(\frac{3}{2}\) the size of the group of squares.

Rate is also an expression of a ratio, but is usually used in more dynamic contexts. Rate is often used to describe a constant relationship in the increase or decrease of measures, such as time or length.

Working with ratio

Working with ratio is part of understanding equivalent fractions and is the essence of proportional reasoning.