The NZ Inland Revenue website offers the following instructions to calculate the GST component of a GST-inclusive amount:

It’s true that this method “will always give you the right answer”, but it’s not obvious why it works or where the numbers 3 and 23 come from. Unpacking and understanding this formula provides a great opportunity to build a better understanding of fractions and the relationship between fractions and percentages.

## What is 15% as a fraction?

Look first at the commonly known relationships between fractions and percentages: 100% = 1 whole, 50% = 1/2 ,25% = ¼, 10% = 1/10

Five percent is half of 10 percent. Now ask your learners (and yourself!) ‘What is half of 1/10?’ You may get a variety of answers from your learners. The correct answer is 1/20. To explain this to your learners, use the number line. One-tenth means divide the distance between 0 and 1 into 10 equal lengths. If we halve the 1/10 lengths, there will be 20 of the halved lengths , each being 1/20 of the full length.

## Calculating the GST component

The bar below represents a GST exclusive price, and has been divided into 20 equal parts, each being 1/20 (or 5%) of the price.

When we add on GST (3/20), the GST inclusive price has 23 equal pieces, each being 1/23 of the GST inclusive price. The GST component is 3/23 of the GST inclusive price.

It should now be apparent that to find the GST component, you divide by 23 (to find the value of one piece) and multiply by 3 (because GST takes up 3 of the 23 pieces). The pre-GST price can be calculated by dividing by 23 and multiplying by 20 -- it's 20/23 of the GST inclusive price.

**Key numeracy outcomes**

As well as making the GST formula meaningful, presenting the GST formula to your learners in this way can build your learners' ability to:

- express commonly used percentages as fractions, so they can use division strategies to calculate percentages, for example find 25% of $80 by dividing by 4
- understand the meaning of the top (numerator) and bottom (denominator) numbers in fractions
- calculate a fraction of a fraction, for example 1/2 of 1/10
- think and reason proportionally, for example if 5% is half of 10%, then its fractional equivalent is half of 1/10.