Research-based guidance and classroom activities for teachers of mathematics

Ratio in mixtures

Ratio in mixtures is a theme within ratio and proportional reasoning. Links to relevant activities and resources are on the right hand side of this page. 

Ratio can be introduced by counting out objects such as ‘five for you, two for me, five for you, two for me ...’ using a model of unequal sharing and correspondence and then comparing the final amounts for you and me.

However, many situations that require an understanding of ratio do not involve countable objects and cannot be visualised as such. Any problems involving mixing liquids, or other continuous quantities such as rice or sand, need better tools than counting, i.e. expressing parts of the whole as fractions or decimals.

When mixing paint we say ‘one part white to four parts blue’. White makes 1/5 of the whole, and there is ¼ as much white as there is blue. Blue makes 4/5 of the whole, and there is 4 times as much blue as white. The relationship between white and blue expressed as a number is ¼ or 4. Expressing visualisable actions as fractions can enable useful reciprocal and additive relations to be deduced, such as ¼ and 4 representing inverse relations, and 1/5 + 4/5 = 1.

Quantities can be measured, but the resulting mixture has a uniform quality which arises from the ratio of different ingredients. One way of looking at this is to think of a mini-recipe, so that a unit portion of the mixture consists of so many grammes of each ingredient; then any overall quantity can be generated by multiplying (scaling-up) the mini-recipe.

Comparing mixtures of different strengths, e.g. the orange drink in jug A is stronger than that in jug B, can be important in helping learners get away from thinking about difference, e.g. jug A contains more orange juice than jug B. It would be possible for one jug to have less juice but a stronger mixture than another.