Multiply just one side of an equation by 1 ? Notation

Question

I have a question about the correct form of notation, this is not about understanding, so I have used 60minutes as an example:

I know that the way to transform minutes to hours is, to multiply them by something that is technically "1":

$ 60min* \frac{1h}{60min} = 1h $

since: $1h=60min |:60min \\ \frac{1h}{60min} = 1$

But what is the correct operation on both sides if I have a equation like:

$ 60min=xh$

If I am not allowed to do any calculus "outside" and I just have to solve it by operating inside the equation - just by equivalence transforming - how would I write that down correctly. Am I allowed to only multiply on one side a value of 1 ?:

$ \quad 60min=xh \\ = 60min* \frac{1h}{60min} = xh \\= 1h=xh \quad|:h \\ \quad x=1 $

Or what would be the correct notation/ writing to solve 60min=xh just by equivalent transformation

Answer 1

Multiplying by $1$ doesn't change the value of anything. Thus if the left hand side and right hand side were equal before you multiplied one of them by $1$, then they're still equal after you've multiplied one of them by $1$ (and, just as important, but more subtle: If they are equal after you've multiplied one of them by $1$, then they were equal before you did so).

The same goes for adding $0$, or simplifying expressions; neither of those operations change the value of anything, so you're allowed to do it to only one side of an equation.

Answer 2

To solve in general such an equation you have to know the relation between hours and minutes in advance. Here you should know that $\text{60 minutes = 1 hour}\quad (1)$.

Now you want to know how many hours 60 minutes are: $\text{60 minutes = x hours} \quad (2)$

With the condition (1) the equation (2) becomes $\text{x hours = 1 hours}\Rightarrow x=1$.

But it makes more sense to calculate the number of hours for different minutes than $60$ minutes, i.e $20$ minutes.

$\text{20 minutes = x hours}$. The condition $\text{60 minutes = 1 hour}$ has to be divided by 3: $\text{20 minutes = 1/3 hours}$. Then the equation becomes

$\text{x hours = 1/3 hours}\Rightarrow x=\frac13$

Answer 3

It is really a bad practice to use unit of measurements inconsistently in an expression. Anyway when this happen you should leave out the indication of the unit of measurements and use instead conversion coefficient. So in keeping with your example you'd better write: $$t_{min} c_{(min\rightarrow h)}=t_{h}$$ You know that $c_{(min\rightarrow h)}=1/60$ because it represents the minutes in an hour. Now your unknown is $t_h$ and you see that the equation is already solved for it: no operations required. So when $t_{min}=60$, $t_h=1$.

If you anyway want to follow your way, you can consider $min$ and $h$ as variable which are linked by such a relation:$$h = 60 min\tag{1}$$

In this case your strange equation: $$60min=xh$$ becomes, after substituting $(1)$ for $h$: $$60min=x *60min$$ that after dividing both sides by $60min$ gives: $$x=1$$