Is it correct to write calculations without units then add them to the answer?

Question

For showing how you solved a problem, is it correct to use the same variable to refer to values both without units and with units? such as:

Q: A farmer wishes to build a fence ... Find the area if 240 meters of fencing are available.

A:

$$ l = \mathrm{length} $$ $$ w = \mathrm{width} $$ $$ ... $$ $$ l = 120 $$ $$ w = 60 $$ $$ A = 120 \times 60 $$ $$ = 7200 \space \mathrm{m^2} $$

Or would you need to name things differently like this:

$$ l = \mathrm{length \space in \space meters} $$ $$ w = \mathrm{width \space in \space meters} $$ $$ ... $$ $$ l = 120 $$ $$ w = 60 $$ $$ A = 120 \times 60 $$ $$ \mathrm{Area} = 7200 \space \mathrm{m^2} $$

For the first style, I have a high school text book which uses it, and so does a math professor: http://web.cs.ucdavis.edu/~amenta/w10/writingman.pdf (p12) but it has the problem that it says $7200 = 7200 \space \mathrm{m^2}$ which I think is technically not true. How can they justify this use of an equals sign for apparently unequal things?

For the second style, it's consistent with how units are used in physics and asked here: https://physics.stackexchange.com/questions/138841/should-i-always-include-units-at-every-step

Clarification: I'm trying to find out if and how the first style is correct, not recommendations for alternative styles such as leaving numbers until the very end or including units all the way through.

Answer 1

It's really useful to keep the dimensions with you along the way, because they provide a check that your answer is in the correct dimensions. If you use some measure wrongly, your units will often tell you - like a parity bit in data transmission. For that reason, I would recommend to always do so.

Technically, the calculation you are making is in units, so it is also correct to retain the units. BUT mathematics is an abstraction of reality, and you could equally argue that you are calculating the abstract, numerical value of the answer and that the units are implicit. So you have a case for removing them but it is disadvantageous to do so for the above reason, so I would keep them at all times.

Answer 2

The two quantities $7200$ and $7200 \;\mathrm{m}^2$ are not the same. One is dimensionless, the other signifies an area.

The convention is to represent the quantity you're working with (regardless of dimension) by symbols when doing calculations and then only at the very last step inserting the numbers including the units.

For instance, say you have a problem of finding some area $A'$ involving a ratio $r=7200$ and a given area $A=7200 \;\mathrm{m}^2$. Then you should do all of your calculations using the notation $r,A,A'$. Let's say you arrive at an expression $A'=rA.$ You should then write $$A'=rA=7200\cdot 7200 \;\mathrm{m}^2=51840000\;\mathrm{m}^2.$$

This approach is good for at least two things:

1. It makes the equations easier to understand and overview and thus makes errors less likely to happen.
2. Looking at the final expression in symbolic form allows one to generalize one's result: If one were to insert all numbers with units from the get go, you'd end up with a mishmash of numbers and units, and so one cannot see how the different quantities grow depending on each other, what the different limits of the problem are, etc., all of which grows your physical intuition for future problems.

Edit: In summary, the first style is not correct, period. The second is technically correct, but is confusing and prone to errors, in my experience.