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Income tax from monopoly
Income tax is not always easy to calculate

Slopes of tax brackets

Some people make a living out of being able to calculate income tax.

In this step we will have a look at how many modern western countries organize their income tax bracket levels. This becomes quite an interesting topic once you join the work force and start pondering where all your tax dollars go.

Australian bank notes in wallet© “Australian bank notes in wallet” by Martin Kingsley/Wikimedia Commons CC BY 2.0

The bracketed nature of income tax

Income Tax is money that you pay the government, and generally the amount you pay is roughly corresponding to how much you make — with some important qualifications.

In most countries Income Tax only kicks in once a person’s income gets to a certain point, and at first has a fixed percentage rate for additional monies earned after that initial point. But then at some higher income, you move into a higher tax bracket, where the slope of the tax/income line increases.

This is a prime example of how important it is for ordinary people to have a basic understanding of linear relationships, and how the geometrical notion of a slope has a tangible, direct and crucial relevance to all of us in our daily lives. Young learners, make sure you understand completely the concepts in this step!

The current story for Australia:

The following rates for 2014-15 apply from 1 July 2014 in Australia. Here is the graph showing yearly gross income on the horizontal \({\normalsize x}\) axis, and income tax on the vertical \({\normalsize y}\) axis. Let’s mark the brackets \(\normalsize{B1}\), \(\normalsize{B2}\), \(\normalsize{B3}\) etc., with corresponding lines \(\normalsize{l1}\), \(\normalsize{l2}\), \(\normalsize{l3}\) etc.

Graph of income tax versus gross income

Q1 (E): What are the slopes of the line segements \(\normalsize{l_2}\), \(\normalsize{l_3}\) and \(\normalsize{l_4}\)? What are the units that these slopes are measured in?

Q2 (M): What would be the equation of the line \(\normalsize{l_2}\) (assuming we extended it)??

Q3 (M): What would be the \(\normalsize{y}\)-intercept of the line segment \(\normalsize{l_2}\) if we extended it to a line?

Taxable income Tax on this income
0 – $18,200 Nil
$18,201 – $37,000 19c for each $1 over $18,200
$37,001 – $80,000 $3,572 plus 32.5c for each $1 over $37,000
$80,001 – $180,000 $17,547 plus 37c for each $1 over $80,000
$180,001 and over $54,547 plus 45c for each $1 over $180,000

Interpreting this table

Please have a good look at this table with a view of understanding precisely what it says. The table in fact has some slopes hidden inside it. Can you see what they are?

For an income less than $18K (here K is a convenient abbreviation for one thousand), there is no tax, so the line \(\normalsize{l_1}\) has slope 0. Between an income of $18.2K and $37K, the table is telling us that the base tax payable is $0.19 for every dollar in Income. This is exactly a slope, since it is telling us that with every increase by $1 in the horizontal direction, there is an increase of $0.19 in the vertical direction. So in this interval, the line \(\normalsize{l_2}\) has slope


For the maximum income in this bracket, namely $37K, the tax payable is thus \(\normalsize{0.19\times (\$37K-18.2K)=\$3572}\). So that number $3572 is not arbitrarily made up–it is a direct consequence of the \(\normalsize{19\%}\) tax rate in this income bracket.

Q4 (M): What would be the difference between earning $80K and $100K in Australia, after tax?

Tax brackets and rates around the world?

It is a very interesting question to ask about income brackets and corresponding tax rates around the world. So here perhaps participants from different countries around the world can tell us: what are the lowest and highest tax rates in your country for individual income taxes? We would expect socialist inclined countries like Sweden to have quite steep tax rates for higher income earners. Other countries with more capitalist orientations will have a flatter tax structure. Where does your country fit in?


A1. The slope of \(\normalsize{l_2}\) is \(\normalsize{0.19}\), that of \(\normalsize{l_3}\) is \(\normalsize{0.325}\), that of \(\normalsize{l_4}\) is \(\normalsize{0.37}\), and that of \(\normalsize{l_5}\) is \(\normalsize{0.45}\). In each case, the slopes are unitless, since they are obtained as ratios of dollars to dollars, so that the dollars cancel out.

A2. The line \(\normalsize{l_2}\) passes through the points \(\normalsize{[18200,0]}\) and \(\normalsize{[37000, 3572]}\) and we know it has slope \(\normalsize{s=0.19}\). So its equation is




A3. The \({\normalsize y}\)-intercept for line \(\normalsize{l_2}\) is \(-0.19 \times 18200 = -3458\).

A4. The difference in the after tax incomes is \(\normalsize{\$12,600}\). The reason is that the extra \(\normalsize{\$20,000}\) attracts a tax rate of \(\normalsize{37\%}\) which amounts to \(\normalsize{\$7400}\).

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Maths for Humans: Linear, Quadratic & Inverse Relations

UNSW Sydney