Addition of different physical quantities We all know the "apples and oranges" rule which says that it's meaningless to add or subtract two different quantities like apples and oranges. But the same rule doesn't hold for the multiplication and division. So my question is, why two quantities with different units cannot be added to one another but the multiplication of them is allowed and how this is expressed mathematically?
 A: I guess I might as well collect my comments into an answer.
You are actually implicitly asking two questions:


*

*whether it makes mathematical sense to add different quantities; and

*if it does make sense why one doesn't encounter it more often in physics.


The answer to 1. is positive in certain cases, especially when talking about fruits. The structure is called free abelian group. It's essentialy one copy of integers for every fruit with addition defined component-wise: $(5a + 2o) + (2a + 3o) = (7a + 5o)$ and so on. One can similarly formalize other concepts of addition of different quantities. One can also introduce multiplication and talk about polymonial rings $K[x,y,\dots,z]$ (where the variables are understood to represent units) or take the field of fractions of that, or even introduce non-commutativity. There are many mathematical structures that can accomodate all of the operations ever needed in physics (and more).
So we come to the point 2. We've seen that it's possible to add different quantities. But that tells you nothing about whether such an operation is ever useful. In particular, when talking about elementary operations used in physical problems to arrive at a result which is always a well-defined quantity with units. I claim that this is why we don't use in physics anything else than addition of quantities with same units because we want to have reasonable units at every step of the calculation.
Note that this is also consistent with taking products of different quantities because this operation doesn't spoil the fact that at every step of the derivation we have a well-defined units of the expression.
A: I think I figured out how this can be expressed mathematically. The units can be thought as mathematical constants and physical quantities as numbers multiplied by that constants (units). So when we add two same quantities with same units we can add the numbers, for example:
$\rm 2\hspace{0.2cm} apples + 5\hspace{0.2cm} apples = 2\cdot a +5\cdot a = (2+5)\cdot a=7\cdot a = 7\hspace{0.2cm} apples $
But for different quantities we can't add the numbers:
$\rm 2\hspace{0.2cm} apples + 5\hspace{0.2cm} oranges = 2\cdot a +5\cdot o $
Therefore,  we can't write:
$\rm 2\hspace{0.2cm} apples + 5\hspace{0.2cm} oranges = 7\hspace{0.2cm} apples + oranges$
However we can multiply (and divide) them, multiplying the numbers and the units:
$\rm 2\hspace{0.2cm} apples \cdot 5\hspace{0.2cm} oranges = (2\cdot a) \cdot (5\cdot o) = (2\ \cdot 5) \cdot (a\cdot o) =10 \cdot (a\cdot o)=10\hspace{0.2cm} apples \cdot oranges$
This seems to work.
A: It's just because the process of multiplying units is well-defined; that's the way with which you define new units. For example, take a square of side $a=1\text{ m}$. To find the area $A$ of the square you square the side, 
$$
A=a^2=(1\text{ m})^2=1 \text{ m}^2.
$$
That defines the square-meter, a unit of area. If you had a rectangle with a side $b=1\text{ m}$ and another side $c=1\text{ ft}$, then you can still find the area $A^\prime$ of the rectangle simply by multiplying $b$ with $c$:
$$
A^\prime=bc=(1\text{ m})\cdot(1\text{ ft})=1\text{ m}\cdot\text{ft}.
$$
This defines a new unit of area, the $\text{m}\cdot\text{ft}$, which is perfectly ok, but not very intuitive to work with.
Adding or subtracting quantities with units is allowed only if the quantities have the same units, simply because in the process of adding or substracting the units are spectators, i.e. they factor out. For example, take a line-segment of length $\ell_1=1\text{ m}$ and a line segment of length $\ell_2=2\text{ m}$. The total length of the two line segments is
$$
\ell=\ell_1+\ell_2=(1\text{ m})+(2\text{ m})=(1+2)\text{ m}=3\text{ m}.
$$
If you couldn't factor out the units, you wouldn't be able to go from step 2 to step 3.
A: In fact, we may add apples and oranges but then we deal with vectors rather than scalars ;-)
I mean $2a + 5o $  is a vector in two-dimensional space with the independent unit vectors $a$ and $o$. It is just (2,5), no problem.
A: A different way of stating the ideas about units is that when an equation is stated in one set of units, we would like to be able to convert that statement into different units. For example, 3 m x 7 m = 21 m2 can be converted into 300 cm x 700 cm = 210000 cm2.
Likewise 3 km + 700 m = 3700 m can be converted into 300000 cm + 70000 cm = 370000 cm.
But there is no sensible way to take 3 gallons + 2 hours = 5 somethings and convert it into, say, a statement in terms of minutes rather than hours.
One highfalutin' way of expressing all this is that we're dealing with affine geometry. When we multiply 3 newtons by 7 meters to get 21 newton-meters, we're finding the area of a rectangle in the affine plane. In affine geometry, distances measured along non-parallel lines cannot be compared, so we can't say that 7 meters is greater than 3 newtons.
A: Multiplication is possible, because you also multiply units of measurement. In other words, if you multiply 2 apples by 3 oranges, what you get is not mere 6, but, 6 orange-apples. Likewise, if you divide 6 apples by 2 oranges, you get 3 apple/orange. This does really make sense as for each orange, there are 3 apples. But when adding or subtracting numbers, you actually can't add or subtract units of measurement.
A: It's not used in Physics AFAIK, at least not in a formal way, but one could write $$5\,\mathrm{apples} + 2\,\mathrm{oranges}=7 (\mathrm{apple\,or\,orange}\!)\!\mathrm{s},$$ or $$5\,\mathrm{men} + 2\,\mathrm{women}=7\,\mathrm{people}.$$ That is, one invents objects that contain other objects. However, I can imagine that there might be inequivalent extensions of the ordinary axioms of arithmetic.
Any Mathematics might in the future be useful in Physics. There is a lot of Mathematics that isn't currently used in Physics, but ruling something out outright seems foolhardy.
A: We count, adimensional. We measure, adimensional. Any measure is a ratio of two quantities: The amount we want to measure is divided by the amount of a chosen standard. To the standard we ascribe a name like meter and regulate a precise way to define it. 
When we measure mass we make a count of barions as found here:
section II. ON UNITS, PHYSICAL LAWS AND SCALING, subsection A. On quantities and units and Atomic measures are number counts
When we add measures we add pure numbers. And the role of the units? Its a remind of what standard was chosen to represent them and the degree of the dimension (Length-L,Time-T,Charge-Q,Mass-M,Temperature). Temperature and Angle measures are pure numbers by definition, not a ratio. As an example $m^3$ has the dimention $L^3$, a volume. It has no meaning the addition of a volume with an area or a temperature with an angle. We can add quantities of the same kind (dimension and degree); a class of fruit or 'object' or 'bag content' allow the counting of both apples and oranges in the same bag. We can add X meters to Y decimeters, the result is still an amount of length. 
Adding quantities of different class, lets say A and B, give a result that belong to a third class (neither A nor B).
Quoting from the above paper: 

"Let us begin by the quantities length and time; length is a
  geometrical, static, concept; time is a concept linked to the ﬂow of
  occurrences, the contrary of static; they are, clearly, distinct
  concepts."

Can we add length quantity to a time quantity? No.
"Base units are not independent", i.e. the length, time, mass and charge units are interdependent; the speed of light and the atom properties interconnect them:  

Time and length units are linked through the speed of light.
  Therefore, while the concepts of length and time are independent,
  their units are not. This has consequences in the description of the
  universe; for instance, relativistic space-time is a property of the
  description of the universe using such units and a reference frame
  calibrated by the method described by Einstein

Multiplication is an addition (just repeated N times) e.g. 5m x 3 = 15m, and 5m x 3m = 15 $m^2$ (dimension $L^2$). Division of quantities of the same nature is a measurement. 
(the author of the above recent paper is a friend of mine and comments are welcome and, if pertinent, I will forward them. Sorry for the bad english.)     
A: Quantities with different units aren't added together because there aren't any physically interesting equations such as $a + b = c$ that allow us to predict a numerical value for one, given the others. On the other hand, there are plenty where you have to multiply values together such as $F = m\frac{dv}{dt},~ C = q/v$. There's also division and forming exponents of quantities.
Adding apples and oranges together is meaningful since it gives the the total number of objects. On the other hand, multiplying them together is generally meaningless because there isn't a meaningful quantitiy this operation calculates a value for.
A: Here is a rhetorical counter-question:
Q: When are 2 feet x 3 feet not 6 square feet (i.e., 6 feet$^2$)? A: When the two measurements were not taken at right angles to one another, and from the same origin point.
In other words, no you cannot multiple apples and oranges.  The units are not related to each other in a defined way.  You can define your own dimensional result, but then you've got something meaningless that you've defined mathematically.  I.e., you can define a word to mean "apples x oranges", but that doesn't mean it has any meaning.
