Perhaps one of your mathematics teachers once told you that "you can't add apples and oranges." Probably this was done to drive home the point that two quantities with different units may not be added together. This isn't a good example, though, because you can add apples and oranges together.
Let's say that you wanted to add
Now, apples and oranges are physical objects, not measurements, so they don't have intrinsic units associated with them. To add unlike things, we just need to find a category broad enough to include both. Instead of calling the apples "apples" and the oranges "oranges," we can call each of them "fruit." If we do so, we can add them together; the answer is
We could go further. If we wanted to add 2 hamburgers to our 3 apples and 4 oranges, the result would be "9 food items." If we add 5 books to that, the result might be "14 things."
If it were true that you can't add apples and oranges, you wouldn't be able to add any real-life objects together, because all objects are different in some way. Say that there are 4 people in your family. You must have got that number by adding them all together. But, if you can't add apples and oranges together because they're different, how could you add all of the people in your family together, when they are all different? If you can't add apples and oranges together, then you can't add (say) one 14 year old boy and one 12 year old girl together. However, you can; one 14 year old boy plus one 12 year old girl equals 2 children. Similarly, one 43 year old male plus one 42 year old female equals 2 adults. Finally, 2 children plus 2 adults equals 4 people.
Note that when it comes to measurements, unlike physical objects, you can't add two measurements together if the units are incommensurable. For example, you can't add 1 metre and 1 kilogram together and get anything meaningful.
See also: The Missing Dollar
Sources used (see bibliography page for titles corresponding to numbers): 62.