Why Two Engineering Calculators Give Different Answers

Two fuel-economy calculators, the same car, the same figures typed in, and answers that differ by several per cent. Neither is broken. They disagree because somewhere upstream each one rounded a definition, and a definition is not a measurement you are allowed to round.

Some numbers are exact by decree

There is a category of number in engineering that looks empirical and is not. Nobody measured how many kilometres are in a mile. In 1959 the national standards bodies of the English-speaking countries agreed the International Yard and Pound Agreement, which defined the yard as exactly 0.9144 metres. Every imperial length follows from that by arithmetic, and the mile comes out at exactly 1.609344 km. Not approximately. The digits stop there because the definition stops there.

Standard gravity is the same kind of object. The value 9.80665 m/s² was adopted by the General Conference on Weights and Measures in 1901 as a conventional reference, and it is exact by definition. It is not the gravitational acceleration where you are standing — that runs from about 9.78 m/s² at the equator to 9.83 at the poles, and drops as you climb — which is precisely why a conventional value was needed. Two engineers in different hemispheres converting kilograms-force to newtons must get the same answer or the drawings do not match.

Where the error gets in

Convert US miles per gallon to litres per 100 km and the conversion factor is 235.215. That number looks arbitrary, and it is why it goes wrong. It is not a constant anybody defined; it is the product of two that were:

100 × 3.785411784 ÷ 1.609344 = 235.2145…

The gallon figure is exact by the same 1959 agreement — the US gallon is 231 cubic inches, and the inch is exactly 25.4 mm. So the whole chain is exact, and the only way to lose accuracy is to round in the middle of it. Use 3.8 and 1.6 and you get 237.5, which is a full one per cent adrift before the car's actual fuel consumption has entered the calculation at all.

That is the general shape of the problem. A rounding error inside a conversion does not stay the size you left it. It multiplies through every subsequent step, and because it is systematic rather than random it never averages out over many calculations — it just shifts every answer the same way, which is far harder to notice than noise.

The practical rule

Carry defined constants at full precision through the whole calculation and round only the answer you are about to report. This costs nothing — the extra digits are free in any calculator or spreadsheet — and it removes an entire class of discrepancy.

The second habit is to state the assumptions alongside the result, because most disagreements between two honest calculations are not arithmetic at all. Air density is the usual culprit: aerodynamic drag goes as density, and density varies with temperature, pressure and altitude by more than any rounding error ever will. A drag figure computed at sea level and 15 °C is a different quantity from one computed at 2,000 m, and quoting the number without the condition makes them look like a contradiction when they are simply two different answers to two different questions.

You can watch that dependence directly rather than take it on trust — the engine efficiency demo and the engine lab let you move the inputs and see which ones actually shift the result.