the logic of why going from “y*x=z” to “z/x=y” is possible.

Question

I ask this in relation to ” /(x/y) ” = ” *(y/x) ”

My mathematical ignorance does not allow me to perceive exactly what it is that confuses me about these manoeuvres and so perhaps my question is vague.

I have no difficulty with it as a technique; as something through which I can put an expression, and out at the other end the right result will appear. What I am trying to understand is *why it works*, contrasted with remembering it as a kind of magical spell.

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**EDIT:**
It was very rewarding for me to read all of your comments. Thank you most kindly for enlightening me.

For those interested in the cause of my previous confusion:
The gaps in my understanding of going from y*x=z to y=z/x were definitions of the equal sign and division.

I can see now that I previously considered the = sign to mean «result» or «answer» in some sort of final sense, like a conclusion; I now see that it only states that this is equal to that.

Following this fundamental piece of knowledge, I can belatedly understand what an equation is. From there, via the definition of division as the opposite of multiplication, I can see that if I divide something while also multiplying it with the same number, these actions cancel each other out.

And so the magical spell between y*x=z and y=z/x is the logic above expressed mathematically as x/(y*x)=z/x.

In: 25

Answer 1

> ” /(x/y) ” = ” *(y/x) ”

If you take 7 minutes to run thousand meters and you want to know how many meters far you can run in 60 minutes you calculate 60 min / (7 min / 1000m) = 8571m OR you can calculate 60 min * (1000m / 7 min) = 8571m.

If you divide 7 by 1000, you get the minutes you need for *one* meter.

If you divide 1000 by 7, you get the meters you can run in *one* minute.

You can think of similar scenarios with buckets of paint that can be used to paint a certain amount of walls.

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I don’t know if that helps anything. It’s not a proof. Do you want a mathematical proof?

Do you accept `x / y = x * (1 / y)`, by the way?

It would be interesting to see your responses to all the answers.