Algebra question (multiple divisions)

[JeenLeen]

If I have x/y/z, does that simplify to something like xz/y?

What if we have four things, like w/x/y/z? Does it simplify to something like wy/xz? (I'm arbitrarily choosing wy/xz here. Just trying to ask what, if anything, it simplifies to so you just have one line of division.)

[eggynack]

The order of operations there is a bit wonky. The functions don't happen simultaneously, and order matters. I think it technically goes left to right, but it makes it quite a bit more straightforward if you write it out explicitly. For x/y/z, if it's (x/y)/z, which I think is the correct reading, then the result is x/yz. For x/(y/z), it would have the result you stated, xz/y. The same is true of w/x/y/z. Something like ((w/x)/y)/z would give w/xyz. To get to your stated result, wy/xz I think it'd have to be like (w/(x/y))/z.

Long story short, parentheses help.

[monomer]

If I have x/y/z, does that simplify to something like xz/y?

What if we have four things, like w/x/y/z? Does it simplify to something like wy/xz? (I'm arbitrarily choosing wy/xz here. Just trying to ask what, if anything, it simplifies to so you just have one line of division.)

You can change the notation of x/y/z to be x/y * 1/z, so you end up with x/yz.
Similarily, w/x/y/z can be written as w/x * 1/y * 1/z, which turns out to be w/xyz

You can test this with regular numbers, say w=2, x=3, y=4, z=5
2/3/4/5 = 0.0333 while 2/(3*4*5)=2/60=0.0333

[Khedrac]

If I have x/y/z, does that simplify to something like xz/y?

What if we have four things, like w/x/y/z? Does it simplify to something like wy/xz? (I'm arbitrarily choosing wy/xz here. Just trying to ask what, if anything, it simplifies to so you just have one line of division.)

With the basic BODMAS rules given equal operators you work from left to right:-

So: x/y/z = (x/y)/z = (x/y) × (1/z) = x/yz

Adding extra terms continues this:

w/x/y/z = (w/x/y)/z = (w/xy)×(1/z) = w/xyz

Etc.

If that's confusing it might be cleared expressed thus:

x/y/z = x × 1/y × 1/z = x/yz

and

w/x/y/x = w × 1/x × 1/y × 1/z = w/xyz

Each time you are adding a new divisor it just gets to multiply the existing divisor.

Edit: doubly swordsaged - and because I posted a double answer each swordsage covered half of it :/

[JeenLeen]

Thanks, y'all. I think that makes sense to me.

I am very much for parenthesis to clarify, but I was given an equation without any parentheses, so I think the default order of operations would be in play.

[eggynack]

Thanks, y'all. I think that makes sense to me.

I am very much for parenthesis to clarify, but I was given an equation without any parentheses, so I think the default order of operations would be in play.
Well, if it was just written out the exact way you wrote it out here, then yeah, w/xyz would be the correct result. However, it's worth asking whether the equation used the implicit parenthesis that is having a bigger division symbol. Such a symbol implies that that is the division which occurs first.

[shawnhcorey]

Thanks, y'all. I think that makes sense to me.

I am very much for parenthesis to clarify, but I was given an equation without any parentheses, so I think the default order of operations would be in play.

Yes, subtraction and division are not commutative. The order in which things appear is important.

[monomer]

Edit: doubly swordsaged - and because I posted a double answer each swordsage covered half of it :/

To be fair, you referenced BODMAS which is a good starting point and something I truthfully forgot to consider (though I know it as PEDMAS; took me a while trying to figure out what BO was before I just went and googled it).

[JeenLeen]

I just found out that a parenthesis was implied, so my problem is actually
(w/x) / (y/z)

I'm sure I could figure that out from the above... but could someone do it quickly?

Well, I'll try, but might get it wrong.
(w/x) / (y/z) = w/x * 1/(y/z)... so w/(x*(y/z))... so w/(xyz)? I feel like I messed up at that last point.

[5a Violista]

I just found out that a parenthesis was implied, so my problem is actually
(w/x) / (y/z)

I'm sure I could figure that out from the above... but could someone do it quickly?

Well, I'll try, but might get it wrong.
(w/x) / (y/z) = w/x * 1/(y/z)... so w/(x*(y/z))... so w/(xyz)? I feel like I messed up at that last point.

(w/x) / (y/z) is the same thing as (w/x) * (1 / (y/z)), yes, but then you messed up on the final step:
It can simplify to (w/x) * (z/y) which then can be written out as w *z /x /y, leading you to (wz)/(xy)


Again,
1/(y/z) is the same thing as (z/y)

[Lord Torath]

I just found out that a parenthesis was implied, so my problem is actually
(w/x) / (y/z)

I'm sure I could figure that out from the above... but could someone do it quickly?

Well, I'll try, but might get it wrong.
(w/x) / (y/z) = w/x * 1/(y/z)... so w/(x*(y/z))... so w/(xyz)? I feel like I messed up at that last point.
You do the parentheses first, and then the external bit. A good way to test this out is to try throwing some integers into your variables and see what results you get:

w=1, x=2, y=3, z=4 => (1/2)/(3/4) = 0.5/0.75 = 0.6667
does 1/(2*3*4) = 2/3? 1/(2*3*4) = 1/24. Nope. So (w/x)/(y/z) =/= w/(xyz)

Also, the definition of dividing a number by a fraction is that you invert that fraction and then multiply the number and the inverted fraction together. So if you've got x/(y/z), you multiply x by z/y. Also note that x = x/1, so you could rewrite that as (x/1)/(y/z).

When dividing a number by a fraction, invert the fraction and then multiply the number and the inverted fraction. Given that, can you see what the answer should be?

Edit: Math-Ninja'd :smalleek:

[5a Violista]

Edit: Math-Ninja'd :smalleek:

You explained it better, though. I was considering explaining about inverting fractions, but I figured showing it might help more - but I'm glad you went into that in your more in-depth answer.

[Jay R]

I can find a way to parse it by blindly applying one of the set of rules for such things, but I cannot guarantee that the person who wrote the equation was using the same approach.

Speaking as a mathematical consultant, the correct answer to this situation is to go back to the source and say, "This is ambiguous. Which of these did you intend?"

[JeenLeen]

I can find a way to parse it by blindly applying one of the set of rules for such things, but I cannot guarantee that the person who wrote the equation was using the same approach.

Speaking as a mathematical consultant, the correct answer to this situation is to go back to the source and say, "This is ambiguous. Which of these did you intend?"

Good point, and that's how I found out there was the implied parentheses.
Also a way I'm pretty sure I got it right now is that i was supposed to do that division to show that a THING is equal to another THING. That division made them look the same, so it should be right (or I did a strange error that, by coincidence, also got the right answer.)

[Knaight]

If I have x/y/z, does that simplify to something like xz/y?

What if we have four things, like w/x/y/z? Does it simplify to something like wy/xz? (I'm arbitrarily choosing wy/xz here. Just trying to ask what, if anything, it simplifies to so you just have one line of division.)

Technically, the really proper way to do this is to read left to right. You start with x, then divide by y, getting xy-1, then divide (xy-1)/z, getting xy-1z-1. This extends indefinitely.

That said, seeing more than one division sign in sequence like that is a sign of garbage notation. As Jay R said, in a real world application you'd check to see what the person who wrote it actually meant. As he didn't say, you'd probably also tell them to use notation that doesn't suck with the absolute minimum allowed diplomacy.


I just found out that a parenthesis was implied, so my problem is actually
(w/x) / (y/z)

That's much better (though the parenthesis really need to be explicit. First off, remember that /x is the same thing as *x-1. That means we have wx-1(yz-1)-1. Exponents multiply, so we have wx-1y-1z. This is (wz)/(yx).

[Algeh]

I just found out that a parenthesis was implied, so my problem is actually
(w/x) / (y/z)




First off, remember that /x is the same thing as *x-1. That means we have wx-1(yz-1)-1. Exponents multiply, so we have wx-1y-1z. This is (wz)/(yx).

Side note, but is converting 1/x to x⁻¹ a common technique that most people are comfortable with as a first thing to try? It's a strategy that I never, ever see my students try (although I see clearly why it would be allowed, it's just not a strategy that examples in our curriculum would use), so it seemed to come out of left field here. (I'm not objecting to it, I'm just used to dealing with people who find exponents really confusing and try to avoid them and so not used to thinking of that approach when it can be avoided.)

I'd probably attack (w/x)/(y/z) by multiplying by (z/y)/(z/y), then dealing with (w/x)*(z/y) and so forth and lots of use of the multiplicative identity to manipulate fractions around, but I'm not sure if this is because so much of my day is spent teaching younger students.

[Knaight]

Side note, but is converting 1/x to x⁻¹ a common technique that most people are comfortable with as a first thing to try?

I have no idea. There's conditions where I use it (e.g. deriving the quotient rule from the product rule if I forget the sign order in the numerator), but "it makes sense to me, therefore it's intuitive" is a line of thought that never goes well.

In the specific context of the horrific notation of this thread though? It intuitively seems like the first thing to do is to get the equation out of that notation. How shared that is is up in the air. I also use exponents really heavily, so there's that.

[jayem]

In the specific context of the horrific notation of this thread though? It intuitively seems like the first thing to do is to get the equation out of that notation. How shared that is is up in the air. I also use exponents really heavily, so there's that.

Context seems good, if it's a ratio of two speeds then the (x1/y1)/(x2/y2) makes more sense (to me) than the equivalent (x1*y2)/(y1*x2) or (x1/x2)/(y1/y2)
and definitely a lot more sense than the totally different x1/(y1*y2*x2), though for something like a volumetric flow x1/(y1*y2*y3) would make more sense (although there are too many exceptions)

Also the units will help, if you've (JeanLean) never come across dimensional analysis it's a fantastic way of sanity checking.
Though note for something like resistivity, where two of the components (area and length) have similar units, it's not so good for deriving. That example also breaks my second rule, as it's all about a volume but area and length act in different options.

[Knaight]

Context seems good, if it's a ratio of two speeds then the (x1/y1)/(x2/y2) makes more sense (to me) than the equivalent (x1*y2)/(y1*x2) or (x1/x2)/(y1/y2)
and definitely a lot more sense than the totally different x1/(y1*y2*x2), though for something like a volumetric flow x1/(y1*y2*y3) would make more sense (although there are too many exceptions).
I can get down with any of these, because you've actually used parentheses; though I'd absolutely reduce (x1/y1)/(x2/y2) to (x1y2)/(y1x2) in basically ayn case. x1/y1/x2/y2 is just terrible notation.



Also the units will help, if you've (JeanLean) never come across dimensional analysis it's a fantastic way of sanity checking.
Though note for something like resistivity, where two of the components (area and length) have similar units, it's not so good for deriving. That example also breaks my second rule, as it's all about a volume but area and length act in different options.
It's an excellent technique in a lot of things, and even if you're using something with weird units (pressure/distance and viscosity/distance for characterizing a tubular reactor of some sort, for instance) it's worth doing.

[BeerMug Paladin]

As others have noted, a/b/c/d would resolve to a/(bcd)

However, sometimes in textbooks, there is sometimes a statement written as

a/b
----
c/d

Where the correct resolution is intended to resolve to the already mentioned (ad)/(bc). I don't know why it's sometimes written like this, since there's no official rule (I'm aware of) to indicate the muchly bigly written division mark is to be resolved after the tinier, tiltier division marks. Still, that's what you get on chalkboards/whiteboards and in textbooks sometimes. I just think that most people pick up on the implied order of operations there without thinking too much about it.

The above statement translated to purely computer text can sometimes be mistakenly written in the first way.

[Douglas]

As others have noted, a/b/c/d would resolve to a/(bcd)

However, sometimes in textbooks, there is sometimes a statement written as

a/b
----
c/d

Where the correct resolution is intended to resolve to the already mentioned (ad)/(bc). I don't know why it's sometimes written like this, since there's no official rule (I'm aware of) to indicate the muchly bigly written division mark is to be resolved after the tinier, tiltier division marks.
A horizontal division mark like that implies parentheses around each of the expressions above and below it, and I think that is a generally recognized standard rule.

[georgie_leech]

A horizontal division mark like that implies parentheses around each of the expressions above and below it, and I think that is a generally recognized standard rule.

It is a fairly straight forward way of denoting "this stuff on top, divided by all this stuff on the bottom."

[BeerMug Paladin]

A horizontal division mark like that implies parentheses around each of the expressions above and below it, and I think that is a generally recognized standard rule.

It is a generally recognized standard rule, but I don't recall it ever being taught formally that way. (Or maybe I just don't recall being taught it.) It's just the kind of thing that is introduced and trivially expected to be adhered to from there on out. That seems okay, since it's introduced sometime in algebra (so theoretically long after you've mastered the concept of order of operations). And the notation is meant to distinguish order of operations for statements like:

a^2 + b
----------
c^2 + d

(Which can include fractional parts for any of the above)

It seems pretty clear that all the stuff on the bottom and all the stuff on the top come before division in that statement. But if one doesn't clearly understand the presumed implication of the above, it might cause some confusion when presented with the same statement involving only fractions for the numerator and denominator. Hence,

a/b
----
c/d

I just thought it was a possible reason for the confusion.