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Okay so I understand that dividing miles by hours gives us "miles-per-hour", with the logic being that we're splitting up a quantity over a group. But what happens if we keep the same dimensions and multiply them?

I understand that multiplication is generally "number of groups" * "quantity inside groups" (i.e. 2 x 3 = 6; 2 groups of 3 is 2, 3's, totaling to 6 )

I also understand if only one set of units were attached it would still make sense (i.e. 1 x 4 inches = 4 inches; 1 group of 4 is 1, 4, totaling 4 inches)

But what happens to the units when both group and quantity have different units? Like so :

1 hour x 5 miles= 5 ?. Clearly 5 belongs there, but what happens to the units? Is this equation saying that there is a relationship between 1 hour and 5 miles? "hour-miles"? What happens to the units? What, if any, is the logic here?

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  • $\begingroup$ Hint example: a light-year is a speed times a time. How do its units come out? $\endgroup$
    – eyeballfrog
    Commented May 25 at 17:12
  • $\begingroup$ @eyeballfrog well, i would say that a new unit is created? "light-year" is a measure of distance, which is neither speed nor time? Edit: If speed is a rate of distance/time, then $\endgroup$
    – ProfessorFinesse
    Commented May 25 at 17:19
  • $\begingroup$ @eyeballfrog Edit: If speed is a rate of distance/time, then light-year = (distance/time) * time, then the relationship is tautological isn't it? Distance being involved means distance is involved. My example has no unit relationships. IDK $\endgroup$
    – ProfessorFinesse
    Commented May 25 at 17:30
  • $\begingroup$ @eyeballfrog okay hang on, 3x * 2y = 6xy. Okay, so 1 hour * 5 cars = 5 car-hours. Okay this is clear, but what is the underlying logic here? Still it has something to do with groups, but i am having a hard time figuring it out. What would a "car-hour" actually be? A new group unto itself? $\endgroup$
    – ProfessorFinesse
    Commented May 25 at 17:33
  • $\begingroup$ Not an answer, but Randall Munroe (author of the celebrated web comic xkcd, as well as books including What If? and How To) might make interesting reading. Particularly, his musings often involve unusual combinations of units, and how to interpret them. For instance, miles per gallon is a reciprocal area (distance divided by volume); One gallon per mile may be viewed as the cross-sectional area of a tube of gasoline one mile long whose volume is one gallon. $\endgroup$
    – Andrew D. Hwang
    Commented May 25 at 18:59

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