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This question already has an answer here:

I totally get what division means in the real world. "dollars / hour", well, that's the number of dollars you will make in one hour. "kilometers / gallon" is the distance you can go with a gallon of gas. Division means that given a certain amount of one thing, you'll get a certain amount of another thing.

I'm so good with division that you could give me a ratio I've never seen before and I can tell you what it means. "Burgers / McDonalds"? It's the average number of burgers a McDonalds will produce. "Dolphin / Miles"? Every mile you drive, you get this amount of dolphins.

Multiplication on the other hand makes no sense at all. What the heck is a foot-pound? And I don't really mean the definition. I mean, what does multiplication actually do for these two values? What does multiplying two values say about their relationship? What if I were to say kilometer-hours? Or dollar-kilograms? Or dolphin-miles? What would those things mean?

Bonus points go to explaining this in a very simple, clear manner.

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marked as duplicate by ACuriousMind, Gert, Kyle Kanos, Sebastian Riese, Norbert Schuch Dec 18 '15 at 16:57

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ It's a decent question, it's just been asked before. You might also be interested in this question. $\endgroup$ – Kyle Kanos Dec 18 '15 at 16:39
  • $\begingroup$ I have a heavy door that needs to be opened. It is a 100 foot-pound door. You can apply a 100 pound force one foot from the hinge, and that will open it. Or a 50 pound force 2 feet from the hinge. Or a 1000 pound force 0.1 feet from the hinge. But if the product of feet-from-hinge and pounds-applied is less than 100, the door will not open. Now does foot-pounds make sense? $\endgroup$ – Eric Lippert Dec 18 '15 at 17:07
  • $\begingroup$ I have a job to do that needs some wire. You are loaning me the wire. It is a 100 meter-hour job. You can loan me 10 meters of wire and I'll give it back in 10 hours. Or you can loan me 20 meters of wire, and I'll give it back in 5 hours. Or you can loan me 1000 meters of wire and I'll give it back in 6 minutes. What this job might actually be, I don't know -- perhaps I am winding a coil and then applying an electromagnetic force for some time. The more wire, the less time I need. Now does meter-hours make sense? $\endgroup$ – Eric Lippert Dec 18 '15 at 17:10
  • $\begingroup$ I build cameras that cost 100 dollar-kilograms. Not 100 dollars per kilogram, but 100 dollar-kilograms. You give me a dollar, I give you a camera that weighs 100 kilograms. You give me 500 dollars, I give you a camera that weighs 0.2 kilograms. Making lightweight cameras is expensive! Now does dollar-kilograms make sense? $\endgroup$ – Eric Lippert Dec 18 '15 at 17:16
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    $\begingroup$ @ericlippert: please don't use comments to answer questions. Since this is a duplicate, you can post your comments as an answer to a nearly identical question and actually gain rep from it. $\endgroup$ – Kyle Kanos Dec 18 '15 at 19:41
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One way to think of different dimensions multiplied together is as a weighting factor that helps transform your unit of measure into something else. I know that this doesn't sound that helpful, but think of the following.

A foot-pound is a unit of torque. It is a measure of 1 pound of force, applied 1 foot away from a pivot point. The distance is a weighting factor that transforms the force you apply into a torque. Larger/smaller distance transform your applied force into larger/smaller torques.

Similar things can be said for other units, such as Newton-meters, which transforms a Force (Newtons) into an energy by weighting it by the distance over which the force is applied. (I realize now that this is the same units as above, but for a different quantity, energy vs torque)

As for some of the more strange units you've listed above, for instance, dolphin-miles, you could use this as a measure of how far dolphins a certain number of dolphins are from a certain point. Adding many dolphin-mile quantities together, and dividing by the total number of dolphins gives you the average position of the dolphins.

You could also use this as a measure of the total distance traveled by a group of dolphins. If 10 dolphins each travel 10 miles, then you would have 100 dolphin-miles of travel. (the same goes for Frisbee's man-hours comment above, which is what reminded me to put this)

Admittedly, things do get weird, because the final unit has to be something that you can make sense of, but this is one way to think of it.

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Man-hours is nonsensical to you? 2 men worked on it for 5 hours is 10 man-hours.

Momentum is mass X velocity

kilowatt hour

kinetic energy is 1/2 m X v X v

Multiplication of two real world values represents exactly that

Kilometer / McDonald makes sense?

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    $\begingroup$ km/McDees makes perfect sense, it's how many kilometers of road are serviced by a single McDee joint! :-) $\endgroup$ – LLlAMnYP Dec 18 '15 at 15:19
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    $\begingroup$ For every McDonald's built, total waistband size increases by that many kilometers. $\endgroup$ – Brian Risk Dec 18 '15 at 15:19
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    $\begingroup$ @Brian Risk Except waistband is the squared cubic root of mass. It's more likely mass increases linearly with the mcdonalds number, so the relationship can't be explained using kilometers/mcdonald. $\endgroup$ – 16807 Dec 18 '15 at 16:15
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    $\begingroup$ Glad we are able to bring clarity to Kilometer / McDonald $\endgroup$ – paparazzo Dec 18 '15 at 16:38
  • $\begingroup$ Kilowatt-hours isn't really a good example here because kilowatts are defined as energy used per unit time. The OP understands that (energy / time ) x time == energy. $\endgroup$ – Eric Lippert Dec 18 '15 at 16:55
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The main point here is that any division is indeed a multiplication. For example, let's use your example of the Burgers $(b)$ and McDonalds $(m)$: \begin{equation} \frac{b_{bu}}{m_{mc}}=b_{bu}\times\frac{1}{m_{mc}}=a_{bu/mc} \end{equation} where $a$ is the average number of Burgers per McDonalds' stores. In this case, what make sense is to multiply the number McDonals's stores with the ratio that was calculated, it is: \begin{equation} m_{mc}\times a_{bu/mc}=b_{bu} \end{equation} which means that if you multiply the number of McDonalds' stores with their average production of burgers, you will obtain the whole production of burgers.

In your question, the following multiplication: \begin{equation} m_{mc}\times b_{bu}=c_{mc\times bu} \end{equation} does not make any sense because the units of the product's result $(mc\times bu)$ does not make any sense in this context. If you were to interpret this result, you would be saying that any McDonald store produces a $b$ amount of burgers (every store produce the overall production!!). That's why you cannot make sense of the product that you stated.

The lesson here is that when doing any kind of math operation, you should never forget about the units of measure of the factors involved as well as the unit of measure of the result. One of the best examples in physics is provided for the gravity acceleration constant $(g)$: \begin{equation} g\approx 9.80_{\frac{m/s}{s}} \end{equation} which means that any falling object in the earth falls in a speed of 9.80 meters per second, but such speed also increases every second (or, in other words, it accelerate while falling).

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