Why does the calculation for freight rate not include division by the freight weight?

Question

I'm doing a little reading about freight rates (the number of miles 1 metric ton can be transported using 1 gallon of diesel fuel). I came across this equation which calculates the freight rate of a trip given its miles traveled and gallons of fuel used:

$Freight\;Rate = \frac{(cargo \; weight \; in \; tons \quad * \quad miles \; traveled)}{gallons}$

The example they provide is:

$Freight\;Rate = \frac{(19 \; tons \quad * \quad 500 \; miles)}{71 \; gallons\;of\;diesel}$

$Freight\;Rate = 1\;ton\;can\;be\;transported\;134\;miles\;using\;1\;gallon\;of\;diesel\;fuel\;$

If final measure states "1 ton can be transported...", then why doesn't the equation involve dividing by the total tonnage of the cargo?

Answer 1

The amount of fuel used is assumed to be proportional to the weight of the cargo. If you double the weight, you double the fuel usage, so the freight rate would remain the same.

Answer 2

Let's just suppose that we can transport $1$ ton of cargo for $134$ miles using $1$ gallon of fuel.

If we spend $2$ gallons of fuel, we could move $2$ tons of cargo for $134$ miles. Or we could take the original $1$ ton and move it $268$ miles.

So it's the same effort to move $2$ tons of cargo for $134$ miles or $1$ ton and move it $268$ miles. Notice that $2 \times 134 = 1 \times 268.$ In other words, the effort required to move cargo for a distance can be measured by

$$ (\text{weight of cargo}) \times (\text{distance moved}). $$

Now back to the original supposition: transporting $1$ ton of cargo for $134$ miles uses $1$ gallon of fuel.

Suppose we want to move $1$ ton of cargo for $500$ miles instead of $134$ miles. Since $500 \approx 134 \times 3.7313,$ we are moving the cargo $3.7314$ times as far so it will use $3.7313$ times as much fuel. That is, $1$ ton for $134$ miles uses $1$ gallon, so $1$ ton for $500$ miles uses $3.7313$ gallons.

But now suppose instead of just $1$ ton moved $500$ miles, we want to move $19$ tons for the same $500$ miles. Now that's $19$ times as much fuel, so we use $19 \times 3.7313 \approx 70.89$ gallons. That rounds to $71$ gallons.

So moving $19$ tons for $500$ miles uses $71$ gallons, which is the same as each $1$ ton moved for $134$ miles used $1$ gallon.

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The key thing here is that we suppose the amount of fuel used is always proportional to the to effort done (weight $\times$ distance), that is, there is a some constant ratio between fuel and effort. Now there are two ways to express a ratio between two quantities $A$ and $B.$ You can write $\frac AB$ or $\frac BA.$ In this case the source you're quoting from has chosen to write the ratio with the effort (weight $\times$ distance) in the numerator and the fuel in the denominator.

One could write it the other way around: fuel in the numerator, effort in the denominator. In that case the original data would look like this:

$$ \frac{71}{19 \times 500} $$

and after simplifying it one might have this:

$$ \frac{1}{1 \times 134}. $$

Now that certainly looks like something got divided by $19.$ But that's just because of the way we chose to write the ratio.

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A more familiar real-life example might be the fuel efficiency of automobiles. In the U.S. this is expressed in miles per gallon. In Europe it's liters per $100$ kilometers. One is distance/fuel, the other is fuel/distance. Both are perfectly legitimate ratios that express fuel efficiency. They just express it in two different ways.