The Fuel Saving That Is Not Linear

Consumption is the reciprocal of economy

There are two ways to describe how thirsty a vehicle is, and they are not two views of the same number. One is distance per unit of fuel — miles per gallon. The other is fuel per unit of distance — gallons per 100 miles, or litres per 100 kilometres. Write the first as E and the second as U and the relationship is U = 1 ÷ E. They carry the same information, but they do not behave the same way under arithmetic, and that is the whole of this article.

What you actually buy is fuel, and what you actually save is fuel. So the quantity that matters over a fixed distance D is the fuel used, D ÷ E. Improving E by a fixed amount does not reduce D ÷ E by a fixed amount, because the derivative of 1/E is −1/E². The saving from one more mile per gallon is proportional to 1 over the square of where you already are. At 10 mpg an extra mpg is worth a hundred times more than at 100 mpg. Nothing about this is subtle once it is written down — but the unit hides it, because the number that gets printed and compared is E, and the human brain reads printed numbers as if they were linear.

The ladder, computed

Fix the distance at 10,000 miles and walk up the scale in equal 10 mpg steps. From 10 to 20 mpg: 1,000 gallons become 500, a saving of 500 gallons. From 20 to 30: 500 become 333, a saving of 167. From 30 to 40: 333 become 250, a saving of 83. From 40 to 50: 50 gallons. From 50 to 60: 33 gallons. The first step is fifteen times the last, and every step is the same size on paper.

Read the second column of the table and the illusion dissolves. In gallons per 100 miles the same five vehicles are 10.00, 5.00, 3.33, 2.50, 2.00 and 1.67 — and now the gaps are 5.00, 1.67, 0.83, 0.50 and 0.33, which are exactly proportional to the fuel saved. That is what a linear unit does: it makes the printed distance between two vehicles equal to the real difference between them. In the reciprocal unit, the printed distance is a distortion that shrinks as the number grows.

Which unit tells the truth

Miles per gallon answers a question nobody asks: how far can I go on a fixed amount of fuel? Almost nobody drives that way. People drive a route, a commute, a delivery round — a distance that is given — and then buy however much fuel it took. The question that matches the behaviour is how much fuel does this distance cost, and that question is answered directly by gallons per 100 miles or litres per 100 kilometres.

The regulators have already conceded the point. The US window sticker prints a gallons-per-100-miles figure alongside the mpg number precisely because the mpg number misleads, and the European label has expressed consumption in litres per 100 kilometres since Directive 1999/94/EC. If you live with mpg, the safest habit is to convert before you compare: divide 100 by the mpg figure to get gallons per 100 miles, then subtract. Do that once and you will never again believe that 38 mpg against 34 is a bigger deal than 18 against 15.

The fleet consequence: fix the worst first

Suppose a small business runs one van and one car, each covering 10,000 miles a year, and has budget to improve exactly one of them. The van does 15 mpg and could be replaced with one doing 18. The car does 30 mpg and could be replaced with one doing 36. The car upgrade is the bigger number in every way a brochure would show it — six extra mpg against three, and it starts from twice the efficiency. Both are the same 20% improvement.

Run the fuel. The van: 10,000 ÷ 15 = 666.7 gallons a year, falling to 10,000 ÷ 18 = 555.6, a saving of 111 gallons. The car: 10,000 ÷ 30 = 333.3 gallons, falling to 10,000 ÷ 36 = 277.8, a saving of 56. The van upgrade saves twice as much fuel as the car upgrade, from half the headline improvement. If the same money buys either, it should buy the van. This is the single most useful thing the reciprocal relationship tells you, and it generalises: in any fleet, the vehicles with the worst consumption figures and the highest annual distances are where the fuel actually is.

Payback is a distance, not a percentage

The same arithmetic decides whether a more efficient car pays for itself. Take an assumed price premium of $3,000 and an assumed pump price of $3.50 a gallon — both illustrative, and both worth replacing with your own numbers. Moving from 30 to 40 mpg saves (1 ÷ 30 − 1 ÷ 40) = 0.008333 gallons per mile, which is $0.0292 per mile. Divide $3,000 by that and the break-even is about 103,000 miles. At 12,000 miles a year that is roughly 8.6 years; at 8,000 miles a year, 12.9 years; at 20,000 miles a year, 5.1 years. Nothing in the brochure told you which of those three you are.

Now do it at the other end of the scale, and the illusion shows its teeth. Moving from 15 to 20 mpg is the same 5 mpg step and a smaller-looking gain, but it saves (1 ÷ 15 − 1 ÷ 20) = 0.01667 gallons per mile, $0.0583, twice as much per mile as the 30-to-40 change. The same $3,000 premium breaks even at about 51,000 miles instead of 103,000. Two purchases, identical price, identical headline improvement in the printed number — and one pays back in half the distance of the other.

Three habits that immunise you

First, always convert to the linear unit before comparing. One division — 100 ÷ mpg — turns the misleading number into an honest one, and it is the same division whichever direction you are going. Second, always fix a distance. A saving quoted with no distance attached is meaningless, because the whole quantity is fuel per distance times distance. Third, put the money last. Convert to fuel first, then to currency, because fuel prices move and the physical saving does not.

The same reasoning runs through the rest of the running-cost arithmetic. Our fuel consumption calculator works in either unit and converts between them; the CO2 emissions article shows that emissions are a fixed multiple of fuel volume, so every gallon on this page is also a fixed mass of carbon dioxide; and the trip cost calculator applies the result to a specific journey rather than to an abstract 10,000 miles.

Frequently asked questions

Is the MPG illusion a real, documented effect?

Yes. Richard Larrick and Jack Soll named and measured it in Science in 2008, showing that people asked to rank fuel-saving choices consistently got them wrong when the figures were given in miles per gallon and got them right when the same information was given as gallons per distance. The US EPA subsequently added a gallons-per-100-miles figure to the fuel economy label. The effect is not about arithmetic ability — the sums are trivial — it is about the unit inviting a linear reading of a non-linear quantity.

Does this mean efficient cars are not worth buying?

No — it means the gain from an efficient car is smaller than the printed numbers suggest once you are already reasonably efficient, and much larger than they suggest when you are not. Going from a very thirsty vehicle to a merely average one is the highest-value fuel decision available. Going from good to excellent is real but modest in fuel terms, and it has to be justified on distance driven, or on something other than fuel — emissions, noise, taxation, or simply wanting the car.

How do I convert between the two units?

Gallons per 100 miles is simply 100 divided by the mpg figure: 25 mpg is 4.00 gal/100 mi, 40 mpg is 2.50. Going the other way is the same operation, because the relationship is its own inverse. If you also need the metric figure, 235.2 divided by mpg gives litres per 100 kilometres, and 235.2 divided by litres per 100 kilometres gives mpg back — that constant is 100 kilometres expressed in miles times a gallon expressed in litres.

Do the official test figures matter, given that nobody achieves them?

They matter as a comparison, not as a prediction. WLTP in Europe and the EPA cycles in the United States are standardised procedures run on a dynamometer, so two cars measured the same way can be ranked against each other with confidence. What they cannot do is tell you your own consumption, which depends on your route, speed, load, tyres, climate and right foot. The right use is to take the official figures for the ranking and your own fill-up records for the absolute level — and the arithmetic in this article works identically on either.

Does the same non-linearity apply to electric vehicles?

It applies to any measure expressed as distance per unit of energy, including miles per kilowatt-hour. The good news is that the electric world mostly settled on the linear unit from the start: kilowatt-hours per 100 miles, or per 100 kilometres, is the figure that appears on labels and in on-board displays, and it subtracts honestly. If you see a range expressed as miles per kWh, convert it — 100 divided by that number — before you compare two cars, exactly as you would for petrol.