Mathematics 11–12 · Year 12

How fast does a car lose value: declining balance depreciation from advertised prices

Depreciation and loans: Depreciation (Mathematics Standard 1, Year 12); Investment and loans: Depreciation (Mathematics Standard 2, Year 12)

Practical, model not builtLow risk

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The idea

A car loses a roughly constant fraction of its value each year, so real advertised prices by age follow a declining-balance curve rather than a straight line.

What you need

At least eight current advertised prices for one make and model with similar kilometres per year, aged 1 to 8 years, collected from a car sales website or a printed used-car price guide
The ATO diminishing value formula: base value x (days held / 365) x (200 per cent / effective life)
A spreadsheet

How to do it

Tabulate age against advertised price and plot the points.
Fit a straight line by eye and state the yearly depreciation D in S = V0 - Dn.
Estimate a declining rate r from two ages with S2 / S1 = (1 - r)^(n2 - n1) and fit S = V0 (1 - r)^n.
Decide which model fits the data better and explain the difference between the two methods.
Compute the ATO diminishing value rate for an asset with an effective life of 8 years and compare it with your fitted r.

What you should see

Worked check with test values: a $40 000 car depreciating at 25 per cent a year is worth $30 000, $22 500, $16 875, $12 656.25 and $9492.19 after 1 to 5 years, while straight-line depreciation of $4000 a year gives $20 000 after 5 years. Test prices of $36 000 at age 1 and $21 000 at age 4 give r = 1 - (21 000 / 36 000)^(1/3) = 16.45 per cent a year. An effective life of 8 years gives the ATO diminishing value rate 200 / 8 = 25 per cent. The learner knows it worked when the declining-balance curve passes closer to the advertised prices than the straight line and the fitted r is stated with the ages used.

What changes

What you change: age of the car
What you measure: advertised price
What you keep the same: same make, model and body type
similar kilometres per year
prices collected in the same week

Common misconceptions

Each of these ideas is wrong, and the activity is a chance to test it.

A car loses the same dollar amount every year; the dollar loss shrinks each year because the rate applies to a smaller value.
A car depreciated at 25 per cent a year is worthless after 4 years; declining balance never reaches zero.

Safety card

Low riskLearners carry it out

Hazards

No hazard is listed.

Controls

No control is listed.

Note

No chemicals and no heat: the NSW Department of Education Chemical Safety in Schools package does not apply. The activity uses published rates and stated test amounts only; no learner's or family's own financial details are needed.

Curriculum references

The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.

Mathematics Standard 11–12 Syllabus (2024), Year 12 Standard 1 focus area Depreciation and loans; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MST-12-S1-03
Mathematics Standard 11–12 Syllabus (2024), Year 12 Standard 2 focus area Investment and loans; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MST-12-S2-02
Australian Curriculum v9No Australian Curriculum v9 code is listed.

Sources

The pages the author read to write this activity.

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