# Can quantities with different units of measure be part of the same system of equations?

For example for this problem:

I used the info in the question to come up with a system of equations:

This system looks reasonably easy to manipulate to find the regular, weekday price. But surely you can't treat quantities of a specific type (e.g. height, weight, price, etc.) as interchangeable with each other in the same way that you would in a problem involving only ordinary numbers?

The answer is apparently 20, but I am not sure how you can get to this solution algebraically.

• This is probably much more info than you need, but there is a fascinating book by John J. Roche, The Mathematics of Measurement (The Athlone Press, London 1998), that details some of the long history of ideas and controversies about what it means to calculate with dimensioned quantities. – Calum Gilhooley Oct 9 at 19:16

When you multiply and divide quantities with units, the units follow suit. In your example, the total spent is in dollars, and the unit price is dollars per kilogram. So the units are $$\frac{\mbox{dollars}} {\mbox{dollars} /\mbox{kilograms}} =\mbox{kilograms}$$ You may be doing this in some other currency if you're using kilograms, but the concept is the same!