1 Equivalence as a concept applied to fractions, decimals and percentages

In mathematics the term ‘equivalence’ is used to mean ‘has the same value’ or ‘can be substituted for’.

Fractions, decimals and percentages each have two main uses:

• Fractions, decimals and percentages are numbers, for example, , 3.2, 78% are all numbers and have a position on a number line
• Fractions, decimals and percentages can be used to denote a proportion of a quantity, for example we can find of £6, 3.2 of 5 kilograms or 78% of 250 people.

Confident learners may be able to move between the number meaning and the proportion meaning of fractions, decimals and percentages but many learners find this confusing.

Consider the following well-known equivalences:

If you think of these as numbers then their equivalence means that they all occupy the same position on a number line. Their equivalence in this case means ‘has the same value’.

If you think of these as proportions then their equivalence means that finding of a given quantity is the same as finding 0.5 of that same given quantity or 50% of that same given quantity. Their equivalence in this case means ‘can be substituted for’.