Which measurement unit is the correct one?

Question

For example,
Speed of car = 60 km/h
Speed of car in 1 hour = 60 km

I know the first one is correct, but how about the second one, and why?

Another example,
Price of 6 bottles = 6 dollars
Price of 6 bottles = 6 dollars/bottle

What about in this example above? What is the relation between the units of word phrase and symbolic phrase: when to write only one unit, and when to write second unit as well?

Another example, from Chemistry,
Mass of 1 mole of carbon 12 = 12 g
Molar mass of carbon 12 = 12 g/mol

Is Molar mass also not saying mass of one mole? Then why different units?

Answer 1

1. instantaneous speed of the car at $(t=1\textrm{ h}) = 60\textrm{km/h}$^҂
   average speed of the car over $1$ hour $= 60\textrm{km/h}$^҂
   total distance travelled by the car in $1$ hour $= 60\textrm{km}$
2. total price of the $6$ bottles $= \$6$
   average price of the $6$ bottles $= \$1$
   unit price of the $6$ bottles $= \$1$
   rate of increase in the total price as quantity bought increases past $12$ bottles $= \$0.80\text{/bottle}$^҂
3. mass of $1$ mole of carbon-$12 = 12\textrm{g}$
   molar mass of carbon-$12$
   $\quad=$ rate of increase in the mass of carbon-$12$ as the number of moles increases
   $\quad=$ mass of carbon-$12$ per mole (from the previous line due to direct proportionality)
   $\quad= 12\textrm{g/mol}$^҂

(All the quantities marked ^҂ are rates; the others are not.)

Answer 2

The speed of the car

It is correct to take "The speed of the car is $60$ km/h" and multiply through by $1$ hour to get "The ____ of the car in $1$ hour is $60$ km". However, it is not correct to call the result "speed". The thing that the car gets $60$ km of in $1$ hour is distance traveled. We can say "The distance the car travels in $1$ hour is $60$ km".

The price of the bottles

If you're buying bottles, you measure the total amount you pay in dollars; what you measure in dollars per bottle is "unit price" (not a terribly standard term). The goal of this "unit price" is to get a property of the thing you want to buy (water or milk or olive oil) that doesn't depend on how much you buy.

In the sentence "The price of $6$ bottles of olive oil is $6$ dollars" and divide through by $6$ bottle to get "The unit price of olive oil is $6$ dollars per $6$ bottles, or $1$ dollar/bottle". Think of this as an equation separated by "is"; you must do the same thing to both sides.

If you divide by $1$ bottle on both sides instead, you still have $6$ left on both sides, and so you have "$6$ times the unit price of olive oil is $6$ dollars/bottle". A true statement, but not a very useful one.

The mass of a mole of carbon

This is a super confusing one.

From one point of view, we're not doing any division. "Molar mass" and "Mass of $1$ mole" are simply synonyms. So we can say "The mass of $1$ mole of Carbon-12 is $12$ grams" or we can say "The molar mass of Carbon-12 is $12$ grams".

From another point of view, we'd like to be able to say things like

100 g of water is about 5.551 mol of water. (Source: Wikipedia)

It sure looks like we have a "mass of water" on one side, and a "number of water molecules" on the other side.

To enable doing this without making mistakes, we write molar mass in units of g/mol, so that we can use the molar mass of a substance to convert from moles to grams. If you want to turn "$1$ mole of Carbon-12" into "$12$ grams of Carbon-12", you multiply by the molar mass of Carbon-12 in grams/mole. That unit is chosen so that the units in $$ 1 \text{ mol} \cdot 12 \text{ } \frac{\text{g}}{\text{mol}} = 12 \text{ g} $$ cancel out.

From a third point of view, a mole is simply a number of particles, so it's unitless(?) anyway.