Real-World Mathematics: Transport, Speed, Travel Time and Fuel

Application of Mathematics in Real-World Usage · Worked application 9 · BTT Mathematics Hub

A route is shorter but takes longer. A vehicle travels the same outward and return distance at different speeds, yet its average speed is not the simple average of those speeds. A more fuel-efficient vehicle does not save the same amount of fuel for every equal increase in miles per gallon. Transport problems are full of quantities that look interchangeable until their denominators are inspected.

This guide follows fictional journeys from distance and time to route choice, fuel use and timetable reliability. All distances, speeds, fuel prices, delays and vehicle efficiencies are invented for Mathematics teaching. They are not live traffic conditions, route recommendations, driving instructions or current transport prices. Real travel decisions should follow applicable laws, signs, official journey information and safe operating procedures.

Start by separating distance, displacement, time and speed

Distance records how much path has been travelled. Time records duration. Average speed is total distance divided by total elapsed time. If a 150 km journey takes 2.5 hours including the time definition chosen for the problem, average speed is 60 km/h.

The formula looks simple because the modelling decision has already been made: which distance belongs to the journey, and what counts inside elapsed time? If a 20-minute stop is excluded in one calculation but included in another, the two reported average speeds answer different questions.

Before calculating, write the boundary. “Moving average speed” excludes stops. “Door-to-door average speed” includes them. A number can be correct under one definition and misleading under another.

Equal distances make the slower leg matter more

Suppose a vehicle travels 60 km outward at 60 km/h and 60 km back at 40 km/h. Outward time is 1 hour. Return time is 1.5 hours. Total distance is 120 km and total time is 2.5 hours, so average speed is 48 km/h.

The simple mean of 60 and 40 is 50 km/h, which is wrong for this journey because the vehicle spends more time at the slower speed. The speeds must be weighted by the time or, more directly, the definition total distance ÷ total time should control the calculation.

For equal outward and return distances travelled at speeds a and b, the round-trip average is 2ab/(a+b), the harmonic mean of the two speeds. The formula is not a trick to memorise; it follows by dividing total distance 2d by total time d/a + d/b.

Equal times create a different average

Now suppose the vehicle travels for one hour at 60 km/h and one hour at 40 km/h. Distances are 60 km and 40 km, so the total is 100 km in two hours. Average speed is 50 km/h.

Here the simple mean works because the time weights are equal. The two examples have the same pair of speeds but different journey structure. The correct average depends on what is held equal.

This is a useful transfer lesson. An “average” is not selected by the visual appearance of two numbers. It is selected by the quantity being accumulated and the denominator defining the rate.

Travel time is a function of distance and speed

Under a constant-speed model, time = distance/speed. A 90 km journey at 75 km/h takes 1.2 hours, or 72 minutes. At 90 km/h it takes one hour. The 15 km/h speed increase reduces the idealised travel time by 12 minutes, not by 20% of the original time.

Time varies inversely with speed when distance is fixed. Doubling speed halves the idealised moving time. Adding the same number of kilometres per hour has a larger time effect at low speeds than at high speeds. This nonlinear relationship is important when interpreting “speed improvements.”

Real journeys include traffic controls, acceleration, congestion, boarding, intersections and other effects, so a constant-speed calculation is a baseline rather than a forecast. State what the model leaves out.

A shorter route is not necessarily a faster route

Consider two fictional routes. Route A is 24 km at an assumed average moving speed of 48 km/h. Route B is 30 km at 75 km/h. Their idealised moving times are 0.5 hour and 0.4 hour respectively: 30 minutes versus 24 minutes.

Route B is six kilometres longer but six minutes faster under the assumptions. If Route B includes an expected 10-minute queue while Route A includes none, the comparison reverses. Route choice is an objective-and-data problem, not a shortest-distance rule.

A useful model records distance, expected time, cost, uncertainty and any hard constraints separately. Combining them into one score before deciding their importance can hide the trade-off.

Timetable slack is the difference between planned and required time

Suppose a journey is planned to arrive at 09:00, and the expected door-to-door duration is 42 minutes. Leaving at 08:10 gives 50 minutes available, so the plan contains 8 minutes of slack.

If a connection requires arrival by 08:55 rather than 09:00, the effective slack falls to 3 minutes. The same journey time has different robustness because the deadline changed.

Slack is not a probability of success. It is a deterministic margin between a plan and a reference duration. To estimate reliability, we also need a distribution or scenario model for actual travel time.

Fuel economy is distance per unit fuel

The U.S. Department of Energy’s Alternative Fuels Data Center glossary defines fuel economy as distance travelled per unit of fuel, commonly miles per gallon in the United States. Its vehicle cost-calculator methodology computes fuel used as distance divided by fuel economy. The same algebra works with any consistent distance-per-fuel unit.

In an invented metric example, a vehicle travels 360 km at 12 km/L. Fuel used is 360/12 = 30 L. At 15 km/L, the same distance requires 24 L. Fuel economy rises by 25%, while fuel used falls by 20%.

The percentages differ because one quantity is the reciprocal of the other for a fixed distance. Increasing kilometres per litre and decreasing litres per kilometre describe the same physical improvement from opposite ratio directions.

Equal gains in fuel economy do not save equal fuel

The DOE’s fuel-economy and consumption data page emphasises that fuel use and miles per gallon are not linearly related. We can see the same Mathematics with invented metric values.

Over 600 km, improving from 6 km/L to 8 km/L reduces fuel use from 100 L to 75 L, saving 25 L. Improving from 18 km/L to 20 km/L reduces fuel use from 33.33 L to 30 L, saving only about 3.33 L. Both improvements add 2 km/L, but they do not have equal fuel consequences.

If the objective is fuel saved, compare litres used over a fixed distance rather than raw increases in the reciprocal measure. This is another reason to identify the decision quantity before choosing a performance metric.

Fuel cost adds price after quantity

Suppose the 30 L in the earlier example is priced at a fictional $2.40/L. Fuel cost is $72. The vehicle using 24 L costs $57.60 at the same fictional price, a difference of $14.40.

If the price changes as well as fuel use, separate the effects. A lower total fuel bill can come from using less fuel, paying less per litre or both. A higher bill can occur despite improved fuel economy if distance travelled or unit price increases enough.

The safest reporting pattern is quantity first, price second: “24 L under the stated efficiency and distance assumptions; $57.60 at the teaching price.” This prevents a monetary result from hiding the underlying resource quantity.

Passenger-kilometres change the denominator again

Suppose Vehicle A travels 100 km using 10 L and carries one passenger. Vehicle B travels the same 100 km using 14 L and carries four passengers. Per vehicle, B uses more fuel. Per passenger-kilometre, A uses 10/(1×100) = 0.10 L per passenger-km, while B uses 14/(4×100) = 0.035 L per passenger-km.

Neither measure automatically answers every policy or personal decision. Vehicle fuel use matters for the actual trip. Passenger-normalised use may matter when comparing occupancy patterns. A denominator can illuminate one question while hiding another.

Always report the total quantity alongside a normalised rate when both are relevant. Normalisation is a comparison tool, not a substitute for the original amount.

A timetable is a sequence of intervals

Consider a fictional trip with three stages: walk 8 minutes, wait 5 minutes, ride 27 minutes. Door-to-door time is 40 minutes. The moving time is 35 minutes only if the wait is the only stationary period and the walking and riding stages are counted as movement.

If the ride covers 18 km and the walk 0.8 km, average moving speed over the 18.8 km and 35 minutes is about 32.23 km/h. Door-to-door average speed over 40 minutes is 28.2 km/h.

The route has not changed. The denominator has. This explains why travel dashboards can legitimately display several time and speed measures without any one being universally “the” speed.

Reliability requires more than the mean

Suppose Route A takes 30, 31, 29, 30 and 30 minutes in five observations. Route B takes 22, 25, 30, 36 and 37 minutes. Both have mean 30 minutes, but their variability differs substantially.

If a hard deadline matters, the distribution of travel time can be more important than the mean. The maximum, upper percentile or probability of exceeding the deadline may be more relevant, depending on the decision.

Five observations are not enough to establish a stable probability model. This is an example of the reasoning developed in Probability, Statistics, Data and Decisions: averages compress evidence, and a decision may depend on the part of the distribution that compression hides.

A weighted timetable can compare service frequency

Imagine a service interval of 6 minutes for two hours and 10 minutes for one hour. A simple average of 6 and 10 is 8 minutes, but the time-weighted mean interval is (6×2 + 10×1)/3 = 7.33 minutes if the question explicitly weights each operating hour equally.

However, average interval alone does not directly equal average passenger wait. Passenger arrivals, bunching and schedule regularity matter. The calculation is useful only at the level of the defined quantity.

This is a recurring applied-Mathematics rule: a summary statistic should not be promoted into a behavioural conclusion unless the assumptions linking them are stated.

Break-even travel comparisons are algebra problems

Suppose Route A takes a fixed 15-minute access and waiting stage plus 1.2 minutes per kilometre. Route B takes a fixed 5-minute access stage plus 1.5 minutes per kilometre. Let d be distance in kilometres.

Set times equal: 15 + 1.2d = 5 + 1.5d. Then 10 = 0.3d, so d = 33.33 km. Below that distance, compare a test value such as d = 10; Route B is faster. Above 33.33 km, Route A becomes faster under the model.

The intersection is a decision boundary, not a prediction that every real journey changes preference at exactly that distance. The coefficients are invented and the real system may be nonlinear or time-dependent.

Uncertainty can be represented by scenarios before it becomes probability

Suppose a route contains 20 km at an average speed somewhere from 40 to 60 km/h plus a waiting period from 5 to 12 minutes. Under the simple independent-range model, moving time ranges from 20/60 hour to 20/40 hour: 20 to 30 minutes. Door-to-door time therefore lies from 25 to 42 minutes.

This interval is a scenario range, not a confidence interval. It assumes the speed range and waiting range capture the relevant possibilities and that the extreme combinations are possible.

If a 35-minute deadline matters, the range crosses it. The Mathematics therefore cannot support a guaranteed on-time statement from these bounds alone. The correct output is a conditional decision: more slack or better evidence is needed if the deadline is hard.

A compact journey model

For one fictional trip, take distance 72 km, average moving speed 60 km/h, planned stop 12 minutes, fuel economy 12 km/L and teaching fuel price $2.50/L. Moving time is 72/60 = 1.2 hours = 72 minutes. Door-to-door time is 84 minutes. Fuel use is 72/12 = 6 L. Fuel cost is $15.

Now change speed to 72 km/h but leave fuel economy at 12 km/L as an explicit simplification. Moving time becomes 60 minutes, door-to-door time 72 minutes, while the modelled fuel quantity remains 6 L. This illustrates a controlled one-variable comparison, not a claim that real fuel economy is independent of speed.

Change fuel economy to 15 km/L while restoring the original 60 km/h speed. Door-to-door time remains 84 minutes; fuel use falls to 4.8 L. The model separates time performance from fuel performance so each change can be interpreted.

Common failure modes

• Simple-mean trap: speeds are averaged without respecting time or distance weights.
• Stop-time ambiguity: one calculation includes waiting while another excludes it.
• Shortest-route assumption: minimum distance is treated as minimum time.
• Reciprocal-rate confusion: km/L and L/km are compared as though they move linearly together.
• Price conflation: a change in money is treated as proof of a change in fuel quantity.
• Normalisation overclaim: per-passenger or per-hour measures are reported without the totals they normalise.
• Average-only reliability: routes with the same mean but different variation are treated as equivalent.
• Model-as-forecast: constant-speed arithmetic is presented as live traffic prediction.

Practice: twenty transport questions

1. Travel 180 km in 3 hours. Find average speed.
2. Travel 40 km at 80 km/h and 40 km at 40 km/h. Find total time and average speed.
3. Travel one hour at 80 km/h and one hour at 40 km/h. Find average speed.
4. Find idealised time for 105 km at 70 km/h.
5. A trip has 50 minutes moving time and 10 minutes waiting. Find door-to-door time.
6. A 24 km route averages 48 km/h; a 30 km route averages 75 km/h. Which has the shorter moving time?
7. A plan allows 55 minutes for a journey expected to take 43 minutes. Find slack.
8. Travel 360 km at 12 km/L. Find fuel used.
9. Travel the same distance at 15 km/L. Find fuel saving relative to question 8.
10. Price 24 L at a fictional $2.30/L.
11. For 600 km, compare fuel use at 6 km/L and 8 km/L.
12. A vehicle uses 12 L for 150 km. Find km/L and L/100 km.
13. Two passengers travel 100 km using 8 L. Find L per passenger-km.
14. A trip has stages 10, 5 and 25 minutes. Find total time.
15. Times are 28, 29, 30, 31 and 32 minutes. Find mean and range.
16. Route A is 15 + 1.2d minutes; Route B is 5 + 1.5d minutes. Find break-even d.
17. A 20 km moving section averages between 40 and 50 km/h. Find its moving-time range.
18. Add a waiting interval of 4–10 minutes to question 17. Find the door-to-door range.
19. A 72 km trip at 60 km/h includes a 12-minute stop. Find door-to-door time.
20. Why does a lower fuel bill not prove better fuel economy?

Worked answers

1. 60 km/h.
2. 1.5 hours total; 53.33 km/h average. The two 40 km legs take 0.5 and 1 hour.
3. 60 km/h. Equal times make the arithmetic mean appropriate.
4. 1.5 hours, or 90 minutes.
5. 60 minutes.
6. The 30 km route: 24 minutes versus 30 minutes.
7. 12 minutes.
8. 30 L.
9. 24 L used; 6 L saved.
10. $55.20. The price is fictional.
11. 100 L versus 75 L; saving 25 L.
12. 12.5 km/L and 8 L/100 km.
13. 0.0004 L per passenger-km. Divide 8 by 2×100.
14. 40 minutes.
15. Mean 30 minutes; range 4 minutes.
16. 33.33 km approximately.
17. 24 to 30 minutes.
18. 28 to 40 minutes. This is a scenario range under the stated independent bounds.
19. 84 minutes.
20. Because cost also depends on price and distance travelled. Compare fuel quantity and a consistent efficiency measure before attributing the cause.

Connected learning and sources

For fuel-economy definitions and the nonlinear relation between economy and consumption, see the U.S. Department of Energy AFDC glossary, vehicle cost-calculator methodology and fuel-economy and consumption page. The numbers on this page are original teaching examples.

Continue to Maps, Bearings, Coordinates and Navigation for spatial route representation; Construction, Area, Volume and Material Estimation for physical quantities; and Inventory, Demand, Reorder Points and Stock Flow for another system where rates and timing determine whether a resource is available when needed. Return to the BTT Mathematics Hub.