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hscscience Maths Std · Y11
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Module 3 · L13 of 14 ~50 min MST-11-04 ⚡ +90 XP available

The Cost of a Vehicle

The number on the windscreen is not what you pay, and what you pay to buy it is not what it costs to keep. Work out the real on-road price, add up a year of running costs, turn that into a figure per kilometre, then build the whole thing as a spreadsheet you can ask questions of.

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Think First

Two cars are for sale. One is $\$18{,}000$ and uses 9 litres of fuel per 100 km. The other is $\$22{,}000$ and uses 6. A friend says the cheaper one is obviously the better buy, because $\$4{,}000$ is a lot of money. Before you calculate anything, write down what you would need to know to answer that properly.

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Learning Intentions

Calculate stamp duty on a vehicle from a given rate table, including above the threshold
Add stamp duty, registration, CTP and transfer fees to a purchase price to get the on-road cost
Tell compulsory from non-compulsory insurance and say what each one covers
Total the ongoing costs of running a vehicle for a year and express them as a cost per kilometre
Model purchasing and running a vehicle in a spreadsheet, and use it to answer a "what if" question
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Key Terms

On-road costEverything you must pay before you can legally drive the car away, not just the advertised price. Like this: a $\$24{,}000$ car with $\$720$ stamp duty, $\$380$ registration, $\$620$ CTP and a $\$40$ transfer fee has an on-road cost of $\$25{,}760$.
Stamp dutyA government tax on the transfer, charged as a rate per $\$100$ of the vehicle's value. Like this: at $\$3$ per $\$100$, a $\$24{,}000$ car has $24000 \div 100 = 240$ lots of $\$100$, so the duty is $240 \times 3 = \$720$.
RegistrationThe annual fee that puts the car on the road and keeps the plates valid. Like this: $\$380$ paid when you buy it, and again every year you keep it, which is why it appears in both lists.
CTP insurance (green slip)Compulsory Third Party insurance, which covers injury to people and must be bought before the car can be registered. Like this: a $\$620$ green slip is not optional, so it belongs in the on-road cost of every vehicle.
Comprehensive insuranceOptional cover for damage to your own car as well as other people's property. Like this: a $\$620$ annual premium with a $\$700$ excess means you pay the first $\$700$ of any claim yourself.
Running costsWhat the car costs you every year after you own it: fuel, servicing, tyres, registration renewal and insurance. Like this: $\$2{,}400$ of fuel plus $\$2{,}100$ of everything else is $\$4{,}500$ a year.
Cost per kilometreAnnual running cost divided by kilometres driven, so two different cars can be compared fairly. Like this: $\$4{,}500$ over $15{,}000$ km is $4500 \div 15000 = \$0.30$, or 30 cents per kilometre.
Fuel consumptionLitres used per 100 km travelled, the figure quoted on every new car. Like this: at 8 L/100 km, driving $15{,}000$ km uses $15000 \times \dfrac{8}{100} = 1{,}200$ litres.
Stamp duty is charged at $\$3$ per $\$100$ of a vehicle's value. How much duty is payable on a $\$24{,}000$ car?
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What It Really Costs To Drive It Away

The advertised price is only the first line of the bill.

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What It Really Costs To Drive It Away

The advertised price is only the first line of the bill. Before the car is legally yours and on the road you also pay stamp duty, registration, a CTP green slip and a small transfer fee. Added together those give the on-road cost.

Stamp duty is the only one you calculate rather than read off a receipt, and exam questions always supply the rate table. A typical one:

  • Up to $\$45{,}000$: $\$3$ for every $\$100$ (or part) of the value.
  • Over $\$45{,}000$: $\$1{,}350$ plus $\$5$ for every $\$100$ (or part) above $\$45{,}000$.

Take a car advertised at $\$24{,}000$. It sits under the threshold, so count the lots of $\$100$ first: $24000 \div 100 = 240$, then charge $\$3$ each: $240 \times 3 = \$720$.

Now add the rest. With registration $\$380$, a CTP green slip $\$620$ and a transfer fee $\$40$:

On-road cost $= 24000 + 720 + 380 + 620 + 40 = \$25{,}760$.

That is $\$1{,}760$ more than the windscreen said, or $\dfrac{1760}{24000} \times 100 = 7.3\%$ (to one decimal place) on top of the advertised price. Worth knowing before you agree to a budget.

Must do: Check which side of the threshold the value falls on before choosing a rate. Above $\$45{,}000$ only the excess is charged at the higher rate, not the whole price.
Common error: Reading "$\$3$ per $\$100$" as 3%. Here they happen to agree, because $\$3$ in every $\$100$ is 3%, but the two-rate table above is not a single percentage and treating it as one gets the over-threshold case wrong.

On-road cost = purchase price + stamp duty + registration + CTP + transfer fee. Stamp duty at 3 per 100: divide the value by 100, then multiply by 3. A 24,000 car: 240 x 3 = 720 duty, on-road 25,760 (dollars).

Pause, copy the on-road cost formula into your book with the $\$24{,}000$ car worked beside it (duty 720, rego 380, CTP 620, transfer 40, total 25,760), and copy the two-line stamp duty table underneath.

A $\$24{,}000$ car carries $\$720$ stamp duty, $\$380$ registration, a $\$620$ green slip and a $\$40$ transfer fee. The on-road cost is $\$$, which is $\$$ more than the advertised price.
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Which Insurance You Must Buy, And Which You Choose

Three kinds of cover turn up in these questions, and the syllabus asks you to tell the compulsory one from the rest.

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Which Insurance You Must Buy, And Which You Choose

Three kinds of cover turn up in these questions, and the syllabus asks you to tell the compulsory one from the rest.

  • CTP, the green slip (compulsory). Covers injury to people in a crash. You cannot register a vehicle without it, so it always belongs in the on-road cost.
  • Third party property (not compulsory). Covers damage you do to someone else's car or property, and nothing of your own.
  • Comprehensive (not compulsory). Covers damage to your own vehicle as well as other people's property. The dearest of the three.

Any of the optional policies can carry an excess: the part of a claim you pay yourself. A $\$620$ premium with a $\$700$ excess means you pay $\$620$ a year, and if you claim for $\$3{,}000$ of damage the insurer pays $3000 - 700 = \$2{,}300$ and you pay the $\$700$.

A lower premium with a higher excess is not automatically cheaper. It is cheaper only if you do not claim, which is a judgement about risk rather than a calculation, and saying so is often the mark.

Must do: Put CTP in the purchase costs, because you cannot register without it, and comprehensive in the running costs, because it is a yearly choice.
Common error: Assuming CTP covers your car. It covers people, not panels. A driver with only a green slip who writes off their own car receives nothing for it.

CTP green slip is compulsory and covers injury to people. Third party property and comprehensive are optional; comprehensive is the only one covering your own vehicle. Excess = the part of a claim you pay yourself, so a 3,000 claim with a 700 excess pays out 2,300 (dollars).

Pause, copy the three insurance types into your book as a short list, writing next to each one whether it is compulsory and exactly whose property it covers.

True or False: CTP green slip insurance is optional, so a careful driver can leave it out of the on-road cost.
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A Year Of Running Costs, And What That Is Per Kilometre

Buying the car is one payment.

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A Year Of Running Costs, And What That Is Per Kilometre

Buying the car is one payment. Keeping it is a payment every year, and the ongoing list is short enough to memorise: fuel, servicing, tyres, registration renewal, insurance.

Fuel is the only one you calculate. Consumption is quoted in litres per 100 km, so:

litres used = distance $\times \dfrac{\text{consumption}}{100}$, then multiply by the price per litre.

Take someone driving $15{,}000$ km a year in a car using 8 L/100 km, with petrol at $\$2.00$ a litre. Litres used $= 15000 \times \dfrac{8}{100} = 1{,}200$ L, so fuel costs $1200 \times 2.00 = \$2{,}400$.

Add servicing $\$700$, tyres $\$400$, registration renewal $\$380$ and insurance $\$620$:

Annual running cost $= 2400 + 700 + 400 + 380 + 620 = \$4{,}500$.

Divide by the distance to get the figure that lets you compare any two cars: $\dfrac{4500}{15000} = \$0.30$ per km, or 30 cents per kilometre.

Notice that only fuel changes when you drive further. Servicing, tyres, registration and insurance are much the same whether you drive $10{,}000$ km or $25{,}000$, so the more you drive, the lower the cost per kilometre becomes. That is why cost per km is only meaningful next to the distance it assumes.

Must do: Divide the consumption by 100 before multiplying. "8 L/100 km" is 0.08 litres per km, not 8.
Common error: Dividing distance by cost instead of cost by distance. $15000 \div 4500 = 3.33$, which is kilometres per dollar, not dollars per kilometre.

Litres used = distance x consumption / 100, then x price per litre. Annual running cost = fuel + servicing + tyres + registration + insurance. Cost per km = annual running cost / distance. 15,000 km at 8 L/100 km and 2.00/L: 1,200 L = 2,400 fuel, 4,500 total, 30 c/km (dollars).

Pause, copy the fuel formula and the cost-per-kilometre formula into your book, with the worked figures beside them: 1,200 L, $\$2{,}400$ fuel, $\$4{,}500$ total, 30 c/km.

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Modelling The Whole Thing In A Spreadsheet

Everything above is arithmetic you could do once on paper.

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Modelling The Whole Thing In A Spreadsheet

Everything above is arithmetic you could do once on paper. The reason to build it as a spreadsheet is that a model answers questions a single calculation cannot: change one number and every total updates.

Lay the running costs out with the inputs in their own cells, so nothing is buried inside a formula:

AB
1Distance per year (km)15000
2Consumption (L/100 km)8
3Fuel price ($\$$/L)2.00
4Fuel cost=B1*B2/100*B3
5Servicing700
6Tyres400
7Registration380
8Insurance620
9Annual running cost=SUM(B4:B8)
10Cost per km=B9/B1

B9 returns $\$4{,}500$ and B10 returns $\$0.30$, the same answers as Card 3. The gain is what happens next: type $25{,}000$ into B1 and the model immediately reports a higher fuel bill but a lower cost per kilometre, because only B4 depends on the distance.

Two rules make the model worth having:

  • Every input gets its own cell. Writing =15000*8/100*2 gives the right number and answers no question, because changing the fuel price means rewriting the formula.
  • Every total is a formula, never a typed number. A typed $4500$ in B9 stops updating the moment anything above it changes, and nothing on screen shows that it has gone stale.
Must do: Use a range in the sum, =SUM(B4:B8), so inserting a row for a new cost is picked up automatically.
Common error: Including B9 in its own sum, or summing B4:B9, which counts the total twice and gives $\$9{,}000$.

Spreadsheet model: every input in its own cell, every total a formula. Fuel =B1*B2/100*B3, annual running cost =SUM(B4:B8), cost per km =B9/B1. Change one input and every total updates, which is what a model gives you that a single calculation does not.

Pause, copy the ten-row layout into your book with the three formulas exactly as written, and note beside B10 why the cost per kilometre falls when B1 rises.

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Worked examples · reveal each step

Worked examples · reveal each step

Worked examples · reveal each step

WE1 · Stamp duty above the threshold. Using the table (up to $\$45{,}000$: $\$3$ per $\$100$; over $\$45{,}000$: $\$1{,}350$ plus $\$5$ per $\$100$ above $\$45{,}000$), find the duty on a $\$52{,}000$ vehicle.
The value is above the threshold, so the first $\$45{,}000$ contributes the flat $\$1{,}350$.
Amount above the threshold $= 52000 - 45000 = \$7{,}000$.
Lots of $\$100$ in that excess $= 7000 \div 100 = 70$, charged at $\$5$ each: $70 \times 5 = \$350$.
Stamp duty $= 1350 + 350 = \mathbf{\$1{,}700}$.
Sense check: charging the whole $\$52{,}000$ at $\$3$ per $\$100$ would give $\$1{,}560$, and the answer must be more than that because the excess is taxed harder ✓.
WE2 · Full on-road cost. A car is advertised at $\$31{,}000$. Stamp duty is $\$3$ per $\$100$, registration is $\$380$, the green slip is $\$645$ and the transfer fee is $\$40$. Find the on-road cost and the extra as a percentage of the advertised price.
Stamp duty $= \dfrac{31000}{100} \times 3 = 310 \times 3 = \$930$.
Extras $= 930 + 380 + 645 + 40 = \$1{,}995$.
On-road cost $= 31000 + 1995 = \mathbf{\$32{,}995}$.
As a percentage of the advertised price: $\dfrac{1995}{31000} \times 100 = \mathbf{6.4\%}$ (1 d.p.).
WE3 · A year of running costs. A car uses 8 L/100 km and is driven $15{,}000$ km a year. Fuel is $\$2.00$ per litre. Servicing is $\$700$, tyres $\$400$, registration $\$380$ and insurance $\$620$. Find the annual running cost and the cost per kilometre.
Litres used $= 15000 \times \dfrac{8}{100} = 1{,}200$ L.
Fuel cost $= 1200 \times 2.00 = \$2{,}400$.
Annual running cost $= 2400 + 700 + 400 + 380 + 620 = \mathbf{\$4{,}500}$.
Cost per kilometre $= \dfrac{4500}{15000} = \$0.30$, that is $\mathbf{30}$ cents per kilometre.
Check the units: dollars divided by kilometres gives dollars per kilometre ✓. The upside-down answer, $15000 \div 4500 = 3.33$, would be kilometres per dollar.
WE4 · Is the cheaper car cheaper? Car A costs $\$18{,}000$ and uses 9 L/100 km. Car B costs $\$22{,}000$ and uses 6 L/100 km. Both are driven $15{,}000$ km a year with fuel at $\$2.00$ per litre. Compare the total cost of buying and fuelling each over 5 years.
Car A fuel per year $= 15000 \times \dfrac{9}{100} = 1{,}350$ L, costing $1350 \times 2.00 = \$2{,}700$.
Car B fuel per year $= 15000 \times \dfrac{6}{100} = 900$ L, costing $900 \times 2.00 = \$1{,}800$.
Over 5 years: Car A fuel $= 5 \times 2700 = \$13{,}500$; Car B fuel $= 5 \times 1800 = \$9{,}000$.
Car A total $= 18000 + 13500 = \$31{,}500$. Car B total $= 22000 + 9000 = \mathbf{\$31{,}000}$.
Car B is $\$500$ cheaper over 5 years, despite costing $\$4{,}000$ more to buy, because it saves $\$900$ of fuel every year and $5 \times 900 = \$4{,}500$ is more than $\$4{,}000$.
The answer depends entirely on the distance and the fuel price. Drive less, or keep the car for fewer years, and Car A wins. Neither car is "obviously" better without the numbers.
A car costs $\$4{,}500$ a year to run and is driven $15{,}000$ km. What is the running cost per kilometre?
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