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
* Start with: y * x = z
* When both sides are equal, you can do anything you want to one side as long as you do the exact same thing to the other side – that way they both *stay* equal.
* Divide both sides by x:
* y * x / x = z / x
* On the left side, the “x / x” is 1 because anything divided by itself is 1
* y * 1 = z / x
* 1 * y is…just y. Because anything times 1 is itself.
* **y = z / x**
When you’re given an equation you’re told that what’s on the left hand side is equal to what’s on the right hand side. So as long as you perform the same operation on both sides, they’ll remain equal. If you subtract 14 on both sides, they’ll still be equal. If you divide by three on both sides, they’ll still be equal. If you take the square root on both sides they’ll remain equal, and so on.
So what you’re doing is really finding operations that makes one side simpler, and then you perform that operation on both sides.
Like if you have `2x + 14 = y` then we’re told that `2x+14` is equal to `y`.
But that means that if we subtract 14 on both sides they’ll still be equal, right? In other words, `2x + 14 – 14 = y – 14`, or simplified a bit, `2x = y – 14`
Next, we can also divide by two on both sides:
`2x / 2 = (y – 14) / 2`, which we can simplify to `x = (y – 14) / 2`. So the “magic spell” as you call it is that we turn the `*` on one side into a `/` on the other side, but what *actually* happened is just that we performed the same division on both sides. On one side it canceled out the `*`, and on the other side, we now have a `/` operation left where there was nothing before.
I have y marbles in each cup, and I have x cups, so I have z marbles all together ( y*x=z ). Now I take all z marbles and divide them out again into x cups. Each cup has y marbles. ( z/x=y )
I have 3 marbles in each cup, and I have 4 cups, so I have 12 marbles all together ( 3*4=12 ). Now I take all 12 marbles and divide them out again into 4 cups. Each cup has 3 marbles. (12/4=3 )
Both “arrangements” of marbles are equally valid. The marbles continue to exist without losing nor gaining any marbles.
>the logic of why going from “y*x=z” to “z/x=y” is possible.
The answer y to the division problem z/x is defined to be the number y such that y*x = z. That is what division *means* period, without any reference to something else. It is defined to be the opposite operation to multiplication. The same is true about subtraction, for the problem z-x=y, y is the number such that y+x = z.
If it helps, addition and multiplication are things you can do on a more basic level, subtraction and division are made up based on analogous notions that they reverse multiplication and addition. Division and subtraction come with problems that you might be familiar with. Examples: Trying to divide 3 hard candies among 5 people, trying to buy 5 apples from a vendor that only has 3. Similar setups for addition and multiplication do not have issues of needing to resort to fractional hard candies or negative numbers of apples to explain away the problem.
>I ask this in relation to ” /(x/y) ” = ” *(y/x) ”
Multiplication (so long as you define fractions and division) admits a type of number called inverses, which is that if I have some number i and some number j that are inverses, then x*i*j = x. The inverse property is that multiplication by j undoes the multiplication by i, and vice versa. It’s a requirement of inverses that i*j = 1. Since x*1 = x, if you do the i*j multiplication first in x*i*j, then you ought to get 1.
Combining these properties, i*j = 1 implies by the definition of division, that 1/j = i. If we bring in some other number k, well then k*1/j = k/j ought to be the same as k*i. So inverses i and j have this property where multiplying by i is the same thing as dividing j.
Turns out the quantity x/y and the quantity y/x are inverses, so you can do this. Multiplying by one is the same as dividing by the other. You can see that they’re inverse by multiplying them together: (x/y)*(y/x) = (x*y)/(y*x) = 1
If you are throwing a party for 10 (x) people and you know that each person eats 3 (y) muffins, how many muffins (z) do you need to buy?
z=x*y=10*3=30
If you have 30 (z) muffins and you want to throw a party to 10 (x) people how many muffins (y) can each person eat?
y =z/x = 30/10=3
Both situations is the same party, the same people and the same muffins, just worded differently
You just need a practical application to see the relationship between the variables. Given the following pie example:
X = number of slices in a pie (number of slices)
Y = weight of the slice (slice weight)
Z = The size of the pie (size of pie)
number of slices * slice weight = weight of pie (X*Y=Z)
alternatively
weight of pie / number of slices = slice weight (Z/X=Y)
or
weight of pie / slice weight = number of slices (Z/Y=X)
ELI5: Anything divided by itself is one, regardless. Because there is an equals sign, x/x = x/x
x^2/x = x on the left hand side of the equation. x/x = 1 on the right side of the equation.
ELI5? You learn this when you are 7. Just wait a little.
Why it works: if you do the same thing to both sides of the = sign, it does not change the equality.
So, in your case, y*x=z, so let’s divide each side by x, then you have y= z/x.
That’s the WHY of why it works, you are doing the same thing to both sides of the equation.
I think this is one I can actually explain to a five-year old! Multiplication is basically a quick way to tally the total of similar groups. Say you have four groups with five people in each: mathematically you would express that as 4 * 5 (4 groups * 5 people) and that equals 20 people in total.
Now, dividing is the opposite of that: You split a large number into smaller groups. So imagine that you have a large group of 20 people, and want to split them into 4 groups. This process would be expressed as 20 / 4 (20 people divided into 4 groups), how many people end up in each group? That’s right, 5 people.
Does that make it clear?
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