Solve fixed and variable vehicle-cost problems

Ontario Grade 11 Mathematics

Personal Finance

Build and use simple cost models to plan vehicle use

Vehicle costs do not all change in the same way. Some costs stay about the same over a chosen period, even if the vehicle is driven more or less. Other costs increase as the vehicle is driven farther. In this lesson, you will sort these costs, represent them with numbers and a simple model, and use the model to answer practical questions. We will use kilometres and Canadian dollars (CAD). A cost model is an estimate, so it is important to state what period and assumptions it uses.

What you will learn

Identify fixed and variable costs in a vehicle-cost situation.
Calculate a variable cost per kilometre from fuel use and fuel price.
Use a simple cost model to find a trip cost or compare two vehicle plans.
Check that a calculated result makes sense in its context.

1. Review: rates, distance, and cost

A rate tells how much one quantity changes for each unit of another quantity. For example, a vehicle that uses 8 litres of fuel per 100 kilometres has a fuel-use rate of 8 L per 100 km. A price of CAD 1.60 per litre is another rate.

To find the fuel cost of a trip, first find how many litres the trip uses, then multiply by the price per litre. If the fuel-use rate is given per 100 km, divide the distance by 100 to find the number of 100 km groups. This approach works for distances that are not exact multiples of 100 as well.

The model in this lesson focuses on costs that can be represented by a fixed amount plus a cost that depends on distance. It does not claim to include every possible vehicle expense. For a fair comparison, use the same time period and include the same kinds of costs for each option.

C = F + r d

Distance is usually measured in kilometres in these examples.
A per-kilometre rate multiplied by kilometres gives a distance-based cost.
State the time period for any fixed cost, such as one year.

2. Sort fixed and variable costs

A fixed cost is a cost that stays the same for the period being considered, even when the distance driven changes. For example, if annual insurance is CAD 1,800, it remains CAD 1,800 in the model whether the vehicle is driven 5,000 km or 15,000 km that year. Actual bills can change over time, but we treat the stated amount as fixed for this calculation.

A variable cost changes with vehicle use. Fuel is a common example: driving more kilometres usually uses more fuel. A variable rate can be written as a cost per kilometre. Other costs that depend on use may be included if the problem gives enough information, but do not assume a rate that has not been provided.

In the model,

C

is the total estimated cost for the period,

F

is the fixed cost for that period,

r

is the variable cost per kilometre, and

d

is the number of kilometres driven. The variable part is r d. When no kilometres are driven, that part is zero, but the fixed cost remains.

A useful first step is to label each given amount. Ask: does this amount stay the same for the chosen period, or does it depend on distance? This prevents adding a fixed annual charge to a trip without recognizing that the two amounts refer to different time spans.

C = F + r d

Fixed cost: stays constant in the model for the selected period.
Variable cost: changes as distance or vehicle use changes.
Total cost combines the fixed amount and the distance-based amount.

3. Find a variable rate from fuel information

Fuel use is often stated in litres per 100 kilometres, while a cost model needs a cost per kilometre. To convert, multiply the litres used per 100 km by the price per litre to get the fuel cost for 100 km. Then divide that cost by 100 to get the cost for 1 km.

For example, at 8 L per 100 km and CAD 1.60 per litre, the fuel cost for 100 km is CAD 12.80. The rate is therefore CAD 0.128 per kilometre. This is an estimate because fuel use and fuel prices can vary.

Keep enough digits during the calculation, then round the final money amount to the nearest cent when appropriate. Rounding the per-kilometre rate too early can create a small difference over a long distance. In a comparison, apply the same rounding approach to both choices.

r = \frac{L}{100}p

Fuel cost per 100 km equals litres per 100 km multiplied by price per litre.
Divide the cost per 100 km by 100 to get the cost per kilometre.
Explain any rounding and remember that the result is an estimate.

4. Use the model for planning and comparison

Once the fixed cost and variable rate are known, substitute the planned distance into the model. The result estimates the total cost for the chosen period. A trip-only question may ask for the variable cost of that trip; include a fixed cost only if the question asks for the full period cost.

To compare two vehicles or plans, calculate each total cost for the same distance and period. One option may have a higher fixed cost but a lower cost per kilometre. The other may be cheaper for shorter distances. Testing distances or finding the distance where the costs are equal helps explain when the comparison changes.

An equal-cost distance is not automatically the best choice. It only shows where the two modeled costs match. The driver should also consider which costs were included and whether the planned distance is realistic for the period.

C_1 = C_2

Match the period and distance when comparing cost models.
Substitute distance into each model and compare the totals.
Check the answer with the original situation and units.

A model’s parts

Part	Meaning	Example
Fixed cost	Stays the same for the chosen period	CAD 2,400 per year
Variable rate	Cost for each kilometre	CAD 0.14 per km
Distance	Kilometres driven in the period	12,000 km
Total modeled cost	Fixed cost plus distance-based cost	CAD 2,400 plus CAD 0.14 per km times distance

Worked example

Estimate the yearly cost for a planned distance

A vehicle has an annual fixed cost of CAD 2,400. It uses 8.4 L of fuel per 100 km, and fuel costs CAD 1.62 per litre. Estimate the total modeled cost for 12,000 km in one year.

Find fuel cost per 100 km
Multiply the fuel use by the price per litre. The litres cancel with the price units, leaving a cost for 100 km.
$8.4(1.62)=13.608$
Convert to cost per kilometre
Divide the cost for 100 km by 100. Keep the unrounded rate for the next calculation to reduce rounding error.
$r=\frac{13.608}{100}=0.13608$
Calculate the distance-based cost
Multiply the rate by the planned 12,000 km. This gives the estimated fuel cost for the year.
$0.13608(12 000)=1 632.96$
Add the fixed cost
The annual fixed cost applies once for the year, so add it to the estimated fuel cost.
$2 400+1 632.96=4 032.96$

Answer: The estimated total modeled cost is CAD 4,032.96 for the year.

Check: The fuel cost is positive and increases with distance. The total is greater than the CAD 2,400 fixed cost, as expected.

Worked example

Find when two vehicle plans cost the same

Plan A has a fixed annual cost of CAD 4,200 and a variable cost of CAD 0.14 per kilometre. Plan B has a fixed annual cost of CAD 2,600 and a variable cost of CAD 0.18 per kilometre. At what annual distance do their modeled costs match, and what is that cost?

Write a model for each plan
Use the fixed amount plus the per-kilometre rate times distance. Use the same distance variable for both plans because the comparison is for the same annual driving distance.
$C_A=4 200+0.14d, C_B=2 600+0.18d$
Set the costs equal
At the distance where the plans cost the same, their total-cost expressions have equal values. Subtracting the smaller fixed amount and combining the distance terms isolates the distance.
$4 200+0.14d=2 600+0.18d$
Solve for the distance
The fixed-cost difference is CAD 1,600. Plan B costs CAD 0.04 more per kilometre, so the extra variable cost reaches CAD 1,600 after 40,000 km.
$1 600=0.04d, d=40 000$
Check the equal total
Substitute 40,000 km into both models. Matching totals confirm the distance.
$C_A=4 200+0.14(40 000)=9 800, C_B=2 600+0.18(40 000)=9 800$

Answer: The plans match at 40,000 km per year. Each modeled cost is CAD 9,800 at that distance.

Check: Below 40,000 km, Plan B's lower fixed cost gives it the lower modeled total. Above 40,000 km, Plan A's lower per-kilometre cost gives it the lower modeled total.

Common mistakes and how to avoid them

Treating the fixed cost as a cost per kilometre.

Correction: Keep the fixed amount separate. Multiply only the variable rate by distance.

Using a fuel rate per 100 km as though it were per kilometre.

Correction: Calculate the cost for 100 km, then divide by 100 before using it as a per-kilometre rate.

Comparing costs from different time periods.

Correction: Make sure both fixed costs cover the same period and both totals use the same distance.

Rounding the rate too early.

Correction: Keep extra digits during the calculation and round the final money amount.

Lesson summary

Fixed costs stay constant in the model for the period being studied.
Variable costs change with distance or vehicle use.
A simple total-cost model adds a fixed cost to a variable rate multiplied by distance.
Convert fuel information to a cost per kilometre before using it in a distance model.
For a fair comparison, use the same period, distance, and included cost types.

Check your understanding

Question 1

A vehicle uses 7 L per 100 km. Fuel costs CAD 1.50 per litre. What is the estimated fuel cost per kilometre?

CAD 0.0105 per km
CAD 0.105 per km
CAD 1.05 per km
CAD 10.50 per km

Show answer and explanation

CAD 0.105 per km

The cost for 100 km is 7 times CAD 1.50, or CAD 10.50. Divide by 100 to get CAD 0.105 per km.

Question 2

A model has a fixed annual cost of CAD 1,900 and a variable rate of CAD 0.12 per kilometre. What is the modeled cost at 5,000 km?

CAD 600
CAD 1,900
CAD 2,500
CAD 7,900

Show answer and explanation

CAD 2,500

The distance-based amount is CAD 0.12 times 5,000, or CAD 600. Add the fixed CAD 1,900 to get CAD 2,500.

Question 3

Two plans have the same fixed cost. Plan A costs CAD 0.16 per km, and Plan B costs CAD 0.20 per km. For a positive distance, which plan has the lower modeled total?

Plan A, because its variable rate is lower.
Plan B, because its variable rate is higher.
They always have equal totals.
There is not enough information to compare them.

Show answer and explanation

Plan A, because its variable rate is lower.

The fixed costs match, so the lower per-kilometre rate gives the lower total for any positive distance.

Key terms

Fixed cost: A cost treated as unchanged for the selected period, regardless of distance driven.
Variable cost: A cost that changes as distance or vehicle use changes.
Variable rate: The variable cost for one unit of distance, such as cost per kilometre.
Cost model: A calculation that estimates total cost using the information and assumptions given.

Continue through MBF3C

View the complete MBF3C Ontario Grade 11 Mathematics curriculum and lessons

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation B3.3. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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