Dividing a given quantity into given ratios

[JeffM]

I am not quite sure what you want me to do with post # 14.

Arithmetic is about learning mechanical procedures and which procedure to use in what situations. Justifying those procedures is frequently more easily done by considering problems using algebra as Dr. Peterson said almost at the start of this thread.

[MATH]a = \text {number of rupees going to Person A.}[/MATH]
[MATH]b = \text {number of rupees going to Person B.}[/MATH]
According to the problem, we are dividing 12 rupees between A and B and the ratio is 4:2.

[MATH]\therefore a + b = 12 \text { and } \dfrac{a}{b} = \dfrac{4}{2}.[/MATH]
That translates the problem into algebraic language.

[MATH]\dfrac{a}{b} = \dfrac{4}{2} \implies a = b * \dfrac{4}{2} \implies 2a = 4b \implies a = 2b.[/MATH]
[MATH]\therefore a + b = 12 \implies 2b + b = 12 \implies 3b = 12 \implies b = 4 \implies a = 8.[/MATH]
Notice that 8/4 = 4/2 and 8 + 4 = 12.

Notice that [MATH]b = 4 \text { because } b = \dfrac{3b}{3} = \dfrac{12}{3} = 2 * \dfrac{12}{6}.[/MATH]
There is no philosophy involved. A very easy problem in algebra can be also be solved in arithmetic by using a procedure that is usually taught by rote because justifying it in general is hard to do without mathematics more advanced than arithmetic. In this case, the procedure is made even more mysterious by failing to state the ratio in lowest terms.

It is, however, possible to show BY ALGEBRA and MATHEMATICAL INDUCTION that the arithmetic procedure will always give the answer easily demonstrated by algebra.

 

[Saumyojit]

In this case, the procedure is made even more mysterious by failing to state the ratio in lowest terms.

My problem was not this.
Yes 2:1 is the simplest form.

If I divide 6 into 12, I get 6 tiny parts of 2 pieces each.

I was saying dividing 12 by 6 will give rs 2 per part.
U said "2 pieces" which sounded ambiguous somewhat.

My first issue with "dividing 12 by 6" was that in the question it was not told that "Divide rs 12 between 6 people" or "How many persons must get rs 12 if each gets Rs 6"-->If this 2 questions were given I would have easily been accepting the fact of dividing 12 by 6.

Here I was saying that till now I knew that I must Divide rs 12 by 6 "only" when the above two questions are given.

 

[JeffM]

My problem was not this.
Yes 2:1 is the simplest form.


I was saying dividing 12 by 6 will give rs 2 per part.
U said "2 pieces" which sounded ambiguous somewhat.
Here I was saying that till now I knew that I must Divide rs 12 by 6 "only" when the above two questions are given.

I was using the word “pieces” because “parts” was being used in too many different senses. There was the part going to A, the part going to B, the six parts that result from dividing 12 by 6, etc.

This whole thing makes clear why we reason with tightly defined symbols rather than slippery words.

 

[Saumyojit]

In short what I understood was that
Parts can mean "of something"(consisting of each value) and Parts can mean "of a whole"(selection from a whole).

We divide 12 by 6 as the question is saying that someone took 4 of something + 2 of something out of 6 parts of Rs 12.

They have given a simpler version of that 2 "Somethings" in the ratio 4:2 .

My HCF method was for me by far the best approach.
End of post.I hope

 

[Saumyojit]

Thanks @Dr.Peterson

What i feel that there is "4 parts of somevalue and 2 parts of another some value " = 6 parts of 12

Just commenting to check my understanding.
4 parts of rs 2 + 2 parts of rs 2 = 6 parts of rs 12.

In "4 parts of rs 2 and 2 parts of rs 2 "
of meant here " consisting of" ; each part is consisting of rs 2.(6 parts of rs 2)

On the rhs side , "6 parts of rs 12 " here 'of' means what each is taken from whole or 12: means I am taking rs 2 six times from 12 .
(or can write Rs 12 divided into 6 parts )

So its the word 'of' Which makes the difference.

And the word part meant "portion of whole".
Divide 1 box into 2 parts ( there will be two parts or segment of 1whole box ) or there are two parts of 1/2 each (here also 2 parts means I am referring to the two portions of the whole).

In a mixture : 1 part of water to 3 parts of wine means a mixture of wine and water of some quantity in litres has been divided into 4 parts .(4 portions)
Where we consider 1 portion.or 1 part (or 1 block just visualising) of the whole (4 parts or 4 blocks) to be filled with water and the other 3 parts or portions or blocks to be filled with wine each block having 1 litre liquid. As every part has to be the same .

Also the unit does not have to fixed but it has to be the same.
Generally we say 1:3 as 1 part of a to 3 parts of b ...if the objects are liquid I can say 1 glass of this to 3 glass of that ...or if the objects are money I can say that 1 box of this to 3 boxes of that ...or buckets, cups etc

 

[HallsofIvy]

To divide a quantity, S, in ratio a:b means to divide it into \(\displaystyle \frac{a}{a+ b}S\) and \(\displaystyle \frac{b}{a+ b}S\) for two reasons.

1) \(\displaystyle \frac{a}{a+ b}+ \frac{b}{a+ b}= \frac{a+ b}{a+ b}= 1\) so \(\displaystyle \frac{a}{a+ b}S+ \frac{b}{a+ b}S= \frac{a+ b}{a+ b}S= S\) so we still have the original amount.

2) \(\displaystyle \frac{\frac{a}{a+ b}}{\frac{b}{a+ b}}= \frac{a}{a+ b}*\frac{a+ b}{b}= \frac{a}{b}\) so we have the right ratio.

 

[Saumyojit]

To divide a quantity, S, in ratio a:b means to divide it into \(\displaystyle \frac{a}{a+ b}S\) and \(\displaystyle \frac{b}{a+ b}S\) for two reasons.

1) \(\displaystyle \frac{a}{a+ b}+ \frac{b}{a+ b}= \frac{a+ b}{a+ b}= 1\) so \(\displaystyle \frac{a}{a+ b}S+ \frac{b}{a+ b}S= \frac{a+ b}{a+ b}S= S\) so we still have the original amount.

2) \(\displaystyle \frac{\frac{a}{a+ b}}{\frac{b}{a+ b}}= \frac{a}{a+ b}*\frac{a+ b}{b}= \frac{a}{b}\) so we have the right ratio.

Another thing I wanted to bring to everyone attention that
Dividing rs 12 into 1.5 parts also gives me rs 8 and rs 4 .
(Partitive division)
12
----- = 8 per 1 part (Quotient is 8)
1.5

After division by any no , Value of Quotient indicates the value of each 1 part .
So originally there are 1.5 parts now 1 part is rs 8 ( as shown above) and the other 0.5 part is :
If 1 part is rs 8 , then 0.5 part amounts to 0.5*8=rs 4 .

Taking the idea from this site

http://mathforum.org/library/drmath/view/77467.html

Only partitive model

Now If I actually represent 12/1.5 in block diagram. Remember to show 12 by a integr is actually easy compared to division by decimal (suppose 12 is a box)

12
------- = 2/3 of 12 per 1 part
(3/2) parts

This 2/3 of 12 is the value of the amount to be taken by first person.

As we can see The quotient is coming 2/3 of the box or of 12 , that means box (12)has to be divided into 3 parts total . |__|__|__|
Now consider the shaded parts or that's 2 parts out of 3 parts

|XX|XX|__|

First two parts of 1/3 or 2/3 (shaded part) of box is the quotient and the remaining 1 part (unshaded part) is 1/3 of 12.

@Dr.Peterson
@JeffM


This sum can be thought in various ways if only one knows so many contexts.
This division by decimal thing came to my mind as I saw one comment of JeffM .
But the visualisation part actually gave me the intuition what was happening behind the scenes . Thanks to drp for that.

And also there are too many usage of parts Which gave me a lesson one English word can have different meanings and came be used in different ways.

So 12 has been divided into 1.5 parts (unequally) where 1 part has rs'8' and the other 0.5 part has rs'4'



1.5 parts of Rs12 =1 part consists of rs 8 and 0.5 part consists of rs 4

Actually I posted this way of my thinking in the original question.(2 months ago)
?
Wow

 

[Dr.Peterson]

Another thing I wanted to bring to everyone attention that
Dividing rs 12 into 1.5 parts also gives me rs 8 and rs 4 .
(Partitive division)
12
----- = 8 per 1 part (Quotient is 8)
1.5

After division by any no , Value of Quotient indicates the value of each 1 part .
So originally there are 1.5 parts now 1 part is rs 8 ( as shown above) and the other 0.5 part is :
If 1 part is rs 8 , then 0.5 part amounts to 0.5*8=rs 4 .

I suppose you could say it that way, but I wouldn't recommend it. "Dividing into 1.5 parts" would make most people just scratch their heads, because we don't talk that way. What you've got here is a model of what "division by 1.5" means in the context of money, so that you can make sense of the nonsense phrase "1.5 parts".

At any rate, please keep in mind that these are only "models" of division, that is, ways in which division might relate to a physical situation. This is not what division actually is. (Division in itself is just the inverse of multiplication.) As you say, applying division (or anything else) to infinitely many different contexts leads to infinitely many models; which is why focusing on them too much leads to confusion.

Just commenting to check my understanding.
4 parts of rs 2 + 2 parts of rs 2 = 6 parts of rs 12.

In "4 parts of rs 2 and 2 parts of rs 2 "
of meant here " consisting of" ; each part is consisting of rs 2.(6 parts of rs 2)

On the rhs side , "6 parts of rs 12 " here 'of' means what each is taken from whole or 12: means I am taking rs 2 six times from 12 .
(or can write Rs 12 divided into 6 parts )

As I've said before, this is just a typical ambiguity in English.

If we said "4 parts of water, 2 parts of wine", or more typically "4 parts water, 2 parts wine" without the "of", we would be referring to equal parts, e.g. 4 liters of water to 2 liters of wine. The word "part" is to be replaced with a unit, or at least thought of as a unit. (A "part" might turn out to be 1.6 liters, for example.)

If we said "divide 12 liters of water into 6 parts", the parts might be anything that adds up to 12, such as 1, 2, 4, and 5 liters.

If we said "divide 12 liters of water into 6 equal parts", each part would be 2 liters.

I would never, ever say "4 parts of rs 2 + 2 parts of rs 2 = 6 parts of rs 12", because I would not want to confuse anyone. If you say it, I have no idea what you mean. Even "4 parts of rs 2" by itself is ambiguous at best. I can't tell whether you are using rs 2 as a total ("parts taken from"), or a part size ("parts consisting of"), or something else. By using "parts of" in two different ways, you are eliminating any context that would normally make it possible to disambiguate the phrases.

You seem to be trying to confuse yourself, by using language in odd ways.

So my answer is, just don't say that! And if you read such things somewhere, show the context to us. If it's written by a non-English speaker, you can just ignore it, because it teaches you nothing.

 

[Saumyojit]

we would be referring to equal parts

Okay generally we refer as equal parts.

We commonly describe, say, a mixture in the ratio of 1:3

Suppose a ratio of 1:3 (cement and gravel) this means 1part cement to 3 parts of gravel or suppose I know that each part is 1kg

So , I can also say this -

1kg cement to 3kg gravel or 1 part of 1 kg cement to 3 parts of 1kg gravel
So here each part is equal .

And I can also check is the ratio 1:3 coming after dividing / simplifying or not?

1 part of 1kg cement

-------------------------------. =1/3=1:3

3 parts of 1kg gravel



But if my ratio is 1:3 then can I write like this,


1 part of 1kg cement

-------------------------------. =1/3=1:3

1 part of 3kg gravel

Here I have kept each part different values but ultimately the ratio is 1:3.
So will I say that in my second case " cement and gravel" is not in a mixture
Something like that??

 

[Dr.Peterson]

1kg cement to 3kg gravel or 1 part of 1 kg cement to 3 parts of 1kg gravel

I don't think the rewording is helpful in understanding; I would never say that. But it doesn't appear to be confusing you at this point, so I won't object.

As I see it, the words "part" and "kg" are equivalent, and we just replace one with the other, rather than putting them together. That is, "part' means "insert unit here".

1 part of 1kg cement
-------------------------------. =1/3=1:3
1 part of 3kg gravel

Here I have kept each part different values but ultimately the ratio is 1:3.
So will I say that in my second case " cement and gravel" is not in a mixture
Something like that??

Here you are switching to a different usage of "part" (no longer "equal parts"), which can only be confusing. If you want to learn the language, use the language as others use it. The difference is not that this is not a mixture (how could that be??), but only that you are using words in a different way than the traditional form for ratios.

 

[Saumyojit]

ratio of water to milk is 1 : 4
​

( for every cup of water there are 4 cups of milk)

Every cup holds same quantity .
Can I say from the diagram that 'x' liter mixture of water & milk is divided into 5 cups . Although when I have something in mixture form, first I have to separate the two quantity and then I can pour in a cup according to the ratios given.
But this I did just for the sake of visualizing "4 parts to 1 part".

I am thinking of another situation : suppose I saw on the table these 5 cups out of which 4 cups of each 1 litre are milk and 1 cup is water . Although each parts are equal; I cannot say there is a mixture of water & milk on the table as they are not mixed but separate .

But if they were in a mixture in the ratio 1:4, i should be saying " 1 part water to 4 parts milk " rather than "1cup water to 4 cups milk" as when in a mixture first i need to separate the two quantity [as long as they are in a mixture i cannot replace the word "part" with "cup " ] and then I can say "1cup water to 4 cups of milk" . I was thinking in terms of real life situation

 

[lev888]

ratio of water to milk is 1 : 4
​

( for every cup of water there are 4 cups of milk)

Every cup holds same quantity .
Can I say from the diagram that 'x' liter mixture of water & milk is divided into 5 cups . Although when I have something in mixture form, first I have to separate the two quantity and then I can pour in a cup according to the ratios given.
But this I did just for the sake of visualizing "4 parts to 1 part".

I am thinking of another situation : suppose I saw on the table these 5 cups out of which 4 cups of each 1 litre are milk and 1 cup is water . Although each parts are equal; I cannot say there is a mixture of water & milk on the table as they are not mixed but separate .

But if they were in a mixture in the ratio 1:4, i should be saying " 1 part water to 4 parts milk " rather than "1cup water to 4 cups milk" as when in a mixture first i need to separate the two quantity [as long as they are in a mixture i cannot replace the word "part" with "cup " ] and then I can say "1cup water to 4 cups of milk" . I was thinking in terms of real life situation

You can say "1 part water to 4 parts milk" if that's the ratio, regardless of whether the components are mixed.

You can say "1 cup water to 4 cups milk" if those are the amounts, regardless of whether the components are mixed.

 

[Saumyojit]

You can say "1 cup water to 4 cups milk" if those are the amounts, regardless of whether the components are mixed

Okay yes it should be amount as When I am using cup each similar size cup holds a specific measure .

regardless of whether the components are mixed

if it is mixed how can I say 1 cup of water ...every cup has water - milk then.

You can say "1 part water to 4 parts milk" if that's the ratio, regardless of whether the components are mixed.

Yes this I agree

 

[Dr.Peterson]

This illustrates the problem of overemphasizing a physical model, as you have been doing.

Can I say from the diagram that 'x' liter mixture of water & milk is divided into 5 cups . Although when I have something in mixture form, first I have to separate the two quantity and then I can pour in a cup according to the ratios given.
But this I did just for the sake of visualizing "4 parts to 1 part".

As shown, it is not a mixture, so you can't call it a mixture! But otherwise, the ratio is 1:4 regardless. If it is really a mixture and you are just illustrating it by the cups, then of course the mixture is 1:4.

am thinking of another situation : suppose I saw on the table these 5 cups out of which 4 cups of each 1 litre are milk and 1 cup is water . Although each parts are equal; I cannot say there is a mixture of water & milk on the table as they are not mixed but separate .

Of course you're right. But that is not a math question!

But if they were in a mixture in the ratio 1:4, i should be saying " 1 part water to 4 parts milk " rather than "1 cup water to 4 cups milk" as when in a mixture first i need to separate the two quantity [as long as they are in a mixture i cannot replace the word "part" with "cup " ] and then I can say "1cup water to 4 cups of milk" . I was thinking in terms of real life situation

Mixed or not, the mixture can still be called 1:4, and still contains 1 cup of water and 4 cups of milk, IF you are using "cup" to refer to the unit, not the container. But in the second question you revealed that you are thinking of cups as containers, by saying each contained a liter; so clearly you can't call it cups in the mixture, since there are no cups in the bowl in which you mixed it!

A mixture shouldn't be described as 1 bottle of water to 1 bottle of milk, since "bottle" does not inherently mean that all the bottles are the same size. But even in terms of units, I would rather not call it "1 liter of water to 4 liters of milk", because that confuses the concept of ratio (which is independent of actual quantities) with a description of the quantities.

 

[Saumyojit]

As shown, it is not a mixture, so you can't call it a mixture! But otherwise, the ratio is 1:4 regardless. If it is really a mixture and you are just illustrating it by the cups, then of course the mixture is 1:4.

ok

Mixed or not, the mixture can still be called 1:4, and still contains 1 cup of water and 4 cups of milk, IF you are using "cup" to refer to the unit, not the container

ok if it was in a mixture , then if i referred ratio as 1 cup of water to 4 cups of milk ;then here by default cup will be the Unit and as I am not mentioning it as a container so i would have no problem .

But in the second question you revealed that you are thinking of cups as containers, by saying each contained a liter; so clearly you can't call it cups in the mixture, since there are no cups in the bowl in which you mixed it!

ok.
So when I am writing like this " 4 cups of 1 litre milk to 1 cup of 1 litre water " here the unit is litre and the cup refer to the container.

would rather not call it "1 liter of water to 4 liters of milk", because that confuses the concept of ratio (which is independent of actual quantities) with a description of the quantities.

seriously i never thought this way .

4 parts of water, 2 parts of wine", or more typically "4 parts water, 2 parts wine" without the "of", we would be referring to equal parts, e.g. 4 liters of water to 2 liters of wine.

there are so many ways to choose .

 

[Dr.Peterson]

So when I am writing like this " 4 cups of 1 litre milk to 1 cup of 1 litre water " here the unit is litre and the cup refer to the container.

Why would any one write that??? You'd just write "4 litres milk to 1 litre water". There is no reason to mention containers.

I think the important thing is not to complicate things. Most of this doesn't really have specific rules, so anything I say may be overgeneralized, and not always apply. But if you avoid using too many words it will often be helpful.

 

[Saumyojit]

Mixed or not, the mixture can still be called 1:4, and still contains 1 cup of water and 4 cups of milk, IF you are using "cup" to refer to the unit, not the container.

So when it is a mixture even if I use cups as long as it is referring to as the unit I have no problem right?
1 cup of water and 4 cups of milk -- generally cups inherently means we are talking about the same size cups right

 

[Dr.Peterson]

So when it is a mixture even if I use cups as long as it is referring to as the unit I have no problem right?
1 cup of water and 4 cups of milk -- generally cups inherently means we are talking about the same size cups right

It isn't the word cup that implies the same size! It's that you are talking about ratios.

 

[Saumyojit]

please see this step by step

1 cup of water and 4 cups of milk

here cup refer to units

Mixed or not, the mixture can still be called 1:4

if it's not mixed then how we can call this a mixture

Now , if there are 4cups of 1litre milk and 1cup of 1litre water separately(not a mixture) as shown in the diagram , then "cups " refer to quantity and litre refer to units .
right ?

A mixture shouldn't be described as 1 bottle of water to 1 bottle of milk

here I think you considered bottle as units so in a ratio each part is same ; so why did you told "bottle" does not inherently mean that all the bottles are the same size .Similarly if I say a mixture of 1cup water to 1cup milk then considering the cup as a unit , then i would not be bothered about their sizes . or (why not in this case i am not worried about the cup sizes )

1 unit of 2.5 liters to 3 units of 2.5 liters

here what is the common unit . It must be litre right ?

 

[Dr.Peterson]

if it's not mixed then how we can call this a mixture

Words can have multiple meanings. Suppose that my "mixed" meant actually stirred together, while "mixture" meant "put together in some way", "combination". Your arguing over words is an utter waste of time.

Now , if there are 4cups of 1litre milk and 1cup of 1litre water separately(not a mixture) as shown in the diagram , then "cups " refer to quantity and litre refer to units .
right ?

Of course not. If I talk about a cup containing a liter of milk, the cup is clearly the container, not a unit, and therefore not a specific quantity. (The unit called "cup" is far less than a liter, anyway, and so is any container I would call a cup.) Please stop talking about cups at all, since that confuses you so much. Replace the word "cup" with "pail" or "bottle" or "bowl" or something, when you are referring to a container; I never used "cup" in that sense except when quoting you. And replace the word "cup" with "litre" when it refers to a unit. Then hopefully you can get back to thinking clearly.

here I think you considered bottle as units so in a ratio each part is same ; so why did you told "bottle" does not inherently mean that all the bottles are the same size .Similarly if I say a mixture of 1cup water to 1cup milk then considering the cup as a unit , then i would not be bothered about their sizes . or (why not in this case i am not worried about the cup sizes )

Once again, context is important. The word "bottle" in general does not refer to a particular size; it is not a standard unit. In a particular statement, we might use it as a "part" with the implication that we are temporarily considering bottles of the same size, or repeated use of the same bottle, but that doesn't make it an actual unit. On the other hand, "cup" is a defined unit in America. My main point was to distinguish between units proper, and containers, because you have been using "cup" in both ways.

here what is the common unit . It must be litre right ?

Are you digging through months of what I've said to find potential inconsistencies? I feel like I'm being persecuted!

In that quote, I was using the word "unit" essentially as we've been using "part". The liter is obviously being used as a unit there, but it is not the "unit" (part) used in the ratio. I was imagining you happen to have some container that holds 2.5 liters, and are using that to measure amounts; so you are treating that as a temporary "unit".

Please stop this silly discussion. I don't have the hours you have to spend on looking through these discussions; you are keeping me from things that matter. I am not going to respond to you further.