Number divided by something divided by something?

[Hammad Shahid]

Homework Statement

For example:
400x/(x/y)

Can also be written as:

400x
____
x
____
ySo now, the answer will be 400y, but my question is, how does “y” get to be multiplied by 400? (Why isn’t it 400 divided by “y”?) How would I undo it for it to not be in the numerator?

 

[David Lewis]

When the denominator gets smaller, the quotient gets larger.
As y increases in size, the denominator gets smaller.

 

[Hammad Shahid]

When the denominator gets smaller, the quotient gets larger.

Hmm... not what I was asking for. Let me see if I can rephrase it.

Here’s an ex of what I’m trying to say: https://imgur.com/a/WhnOo

 

[jedishrfu]

You need to pay attention to the parentheses.

You have 400x divided by (x/y) so the x’s cancel leaving 400 / (1/y) which is the same as 400 * y

How do you get that answer by multiplying top and bottom by y.

400 * y      400*y
————————— = ————-
(1/y) * y      1

 

[David Lewis]

It's 4000J / 100J / 1^oC

So in the denominator, we have 100J / 1^oC (=100)

 

[jedishrfu]

It's 4000J / 100J / 1^oC

So in the denominator, we have 100J / 1⁰C (=100)

You changed the problem by removing the parentheses to be (4000j / 100j) / 1c

I think the 100j per 1c is more of a unit of measure so that your answer will be in degrees c.

 

[jedishrfu]

An analogous problem would be I travel for 100 kilometers at 25 km/hr hence it takes me 4 hours to make the journey.

 

[Mark44]

For example:
400x/(x/y)

So now, the answer will be 400y, but my question is, how does “y” get to be multiplied by 400? (Why isn’t it 400 divided by “y”?)

Because dividing by a/b is defined to be the same as (i.e., equal to) multiplication by the reciprical of a/b. In other words, for your problem,
$\frac {400x}{\frac x y} = 400x \cdot \frac y x = \frac {400x y} x = 400y$

 

[Hammad Shahid]

I see, thank you.
I don’t think I was ever really shown that in school, so I never thought about it that way. Is there a guide for practice/example of how this works?

 

[Hammad Shahid]

Also, so this means if a problem is x/y/z , you cannot do (x/y)/z, right?

 

[Hammad Shahid]

Oh I remember, it’s that one thing I learned from fifth grade I think. It’s when you divide a fraction by a fraction, it’s the same as multiplying the first fraction by the reciprocal of the second.
So in this case, if it’s 400x/x/y, we could put it as (400x/1) / (x/y), and rewrite it as (400x/1)*(y/x). Am I write about this?

 

[Mark44]

Also, so this means if a problem is x/y/z , you cannot do (x/y)/z, right?

Yes, you can do this. By the usual order of operations, and since both operations are division, you can do the first division, and then divide that by z.

So x/y/z is the same as
$$\frac{\frac x y}{z} = \frac x y \cdot \frac 1 z = \frac {x y}z$$

Edit: The last expression should be $\frac x {yz}$

Oh I remember, it’s that one thing I learned from fifth grade I think. It’s when you divide a fraction by a fraction, it’s the same as multiplying the first fraction by the reciprocal of the second.

Yes, that's right.

So in this case, if it’s 400x/x/y, we could put it as (400x/1) / (x/y), and rewrite it as (400x/1)*(y/x). Am I write about this?

Yes, you are right. Note that "write' is an action you do with a pen or pencil.

 

[Ray Vickson]

Homework Statement

For example:
400x/(x/y)

Can also be written as:

400x
____
x
____
ySo now, the answer will be 400y, but my question is, how does “y” get to be multiplied by 400? (Why isn’t it 400 divided by “y”?) How would I undo it for it to not be in the numerator?

Using LaTeX makes it clearer: your original expression $400\, x /(x/y)$ can be displayed as
$$\frac{400\, x}{ \displaystyle \frac{x}{y}} $$
This can also be written as
$$\frac{400 \,x}{x} \times \frac{1}{\displaystyle \frac{1}{y}} $$
or more clearly as
$$\frac{400 \,x}{x} \times \frac{1}{\displaystyle \left(\frac{1}{y}\right)} ,$$
and
$$\frac{1}{\displaystyle \left(\frac{1}{y}\right)} = y$$

 

[Ray Vickson]

Yes, you can do this. By the usual order of operations, and since both operations are division, you can do the first division, and then divide that by z.

So x/y/z is the same as
$$\frac{\frac x y}{z} = \frac x y \cdot \frac 1 z = \frac {x y}z$$

**********************
Oops:
$$ \frac{x}{y} \cdot \frac{1}{z} = \frac{x}{yz} \;\;!$$

The Classic Maple interface accepts "a/b/c" instead of "a/(b*c)"

***********************

Yes, that's right.
Yes, you are right. Note that "write' is an action you do with a pen or pencil.

 

[Mark44]

**********************
Oops:
$$ \frac{x}{y} \cdot \frac{1}{z} = \frac{x}{yz} \;\;!$$

Thanks for catching that, Ray. I have edited my earlier post.

The Classic Maple interface accepts "a/b/c" instead of "a/(b*c)"

Are you saying that Classic Maple treats a/b/c as if it were written a/(b * c)?
That's odd, since programming languages generally associate arithmetic operators at the same precedence from left to right, with a + b + c being interpreted as if written (a + b) + c, and a / b / c as if written (a /b) /c.

 

[Ray Vickson]

Thanks for catching that, Ray. I have edited my earlier post.
Are you saying that Classic Maple treats a/b/c as if it were written a/(b * c)?
That's odd, since programming languages generally associate arithmetic operators at the same precedence from left to right, with a + b + c being interpreted as if written (a + b) + c, and a / b / c as if written (a /b) /c.

Saying "a/b/c = a/(b*c)" is not odd at all; it is completely consistent with the usual priority rules, because (a/b)/c = a/(b*c).

However, the new Maple interface does not do that, because it immediately converts a typed a/b to $\frac a b$; in other words, it would not even let a user write "a/b" on screen, and so would not even allow one to type in a/b/c on-screen---it would convert fractions immediately in real time to
$$\frac{a}{\frac{b}{c}}$$
However, a/b/c copied from a classical worksheet and pasted into the new interface would appear on-screen as "a/b/c" and would still evaluate as $\frac{a}{bc}$, just the way it did in the classic interface. To get a/(b*c) in the new interface, one types a/b*c, and to get (a/b)*c one types a/b→*c, where → is the right-arrow key. The → key closes out a fraction or a power, putting one back on the previous input level. BTW: the new interface can be quite handy when one get used to it: entering $a/(b+c)$ is easy: just type a/b+c, because it starts off as $\frac{a}{b \;\;\;}$, then waits for more input from the user. If one does not leave the fraction (by prsssing the →) then when one types the "+c" that immediately goes to the right of the "b" in the denominator. If one actually wants $\frac{a}{b}+c$ it needs "a/b →+c", which can be a bit annoying.

 

[Mark44]

Saying "a/b/c = a/(b*c)" is not odd at all

I misspoke. It's early in the morning for me, and my brain isn't operating at full capacity yet. I must have been thinking of something like a/b*c, where some might interpret this to mean $\frac a {b \cdot c}$ and others as $\frac a b \cdot c$.

The PEDMAS precedence rules aren't as clear as the precedence rules in programming languages, particularly C and languages derived from C. The programming precedence rules specify not only the precedence of all the operators, but also how the operators associate or group, for expressions that include multiple operators of of the same precedence level.

A mathematical expression such as a/b*c is ambiguous, based on several threads posted here in the past, but the same expression is crystal clear if parsed by a C or C++ compiler. Since the division and multiplication operators are at the same precedence level, and associate left-to-right, a C compiler would interpret a/b*c as if written (a/b) * c.