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.