# Dividing a given quantity into given ratios

#### Saumyojit

##### Full Member
DIVIDE Rs 12 into 4:2 between 2 persons .
In most of the solutions they are doing like this

"They are adding the ratio terms 4 and 2 getting 6 by which 12 will be divided
Rs12/6parts=2Rs/ part .
Now they say we need to take 4 of rs2 and 2 of rs2
So the shares are rs 8 and rs 4 "

Now after seeing the soln what I feel is that I have

4 parts consists of rs 8 and 2 parts consists of rs 4 = 6 parts of Rs12
or
4 groups of rs 2 and 2 groups of rs 2 = 6 groups of Rs 2

{Parts mean Division , don't think because I wrote parts of 12 that means 'of' means multiplication so I multiply 6*12 ;! no that is wrong ..
I can write 1/6 of 12 or 6 parts of 12
Groups & parts are different}

That means the given ratio 4:2 in the question was a simpler form of x:y where X represents the share(the money out of 12) of first person and y (the remaining money out of 12) of the second person.

Q1: Why did they add 4 +2? THE RATIO TERMS . Thats coz we want to find the size of each 1 part?

Q2: What does 4 : 2 actually represent? Is it just like i said above [ given ratio 4:2 in the question was a simpler form of x:y where X represents the share(the money out of 12) of first person and y (the remaining money out of 12) of the second person.]
OR
Represents the ratio between " How many GROUPS of each part value taken by the first person : second "

Q3: WHY DO WE DIVIDE 12 by 6?

CONFUSION: if the question was given like this "Find the share of each person if 12 is divided among 6 persons" then i can or should then , divide 12 into 6 parts or persons
But there are two terms in a ratio that means there are two persons not 6! -Thats the trick or where my confusion arises
Now if i know there are two persons also
I cannot do rs12/2persons as that will mean I am dividing rs 12 in 1:1 or 2:2 or 3:3
But they have given 4:2 clearly (unequal share)
--------------------------------------------------------------------END OF FIRST PART -----------------------------------------------------------------

Now acc to solution I am dividing rs 12/6
Then getting the answer rs 4 and rs 8
I can make one expression by asking Rs'12' has been divided into how many parts so that one of the part is rs '8'
12/X=8
X=3/2
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

One thing i have noticed that "Ratio between the parts is proportional to ratio between the sizes"
(1 part / 0.5 part ) = (Rs 8 /Rs 4)

I can say : Rs 12 is divided into rs 4 and rs 8 OR Rs 12 is divided into 1.5 parts

SO if 12 is divided into 1.5 parts then also ; the person A is getting Rs '8' and the person b is getting Rs '4' (the amount receiving is same) . But the ratio then would become 1 : 0.5 . which is VIOLATING the original given ratio 4:2

So 12 has been divided into how many parts originally then 6 or 1.5 ? I am getting confused .

#### Dr.Peterson

##### Elite Member
DIVIDE Rs 12 into 4:2 between 2 persons .
In most of the solutions they are doing like this

"They are adding the ratio terms 4 and 2 getting 6 by which 12 will be divided
Rs12/6parts=2Rs/ part .
Now they say we need to take 4 of rs2 and 2 of rs2
So the shares are rs 8 and rs 4 "
You're making this far too complicated, in part by trying to make it more concrete than it has to be. I'm not going to bother going through your specific questions.

You want to split Rs 12 into two parts we can call 4k and 2k, so that the ratio between them is 4k:2k = 4:2. (I don't know why they didn't call it 2:1!)

So we need 4k + 2k = 12. This simplifies to 6k = 12, and clearly k = 2.

So the parts are 4(2) = Rs 8 and 2(2) = Rs 4.

We added 4+2 because we wanted a sum. We divide by 6 because we want to find k.

If you want to keep in concrete, picture actual rupee coins or bills. You have 12 of them. To make a ratio of 4:2, you can just put 4 into one pile and 2 into another. That uses up Rs 6. You have another 6, so you repeat; now you have Rs 8 in one pile and Rs 4 in the other.

Observe that adding 4+2 arises naturally in keeping track of how much is used up in each round; and although we didn't actually divide by 12, we could have done so to find how many rounds we needed.

Each of these (and other things I could have done instead) are just particular ways to solve the problem; each uses a different representation. You don't need to analyze the individual "objects", whether they are "groups" or "parts" or whatever; those are just models of the abstract ideas underlying the problem.

#### Saumyojit

##### Full Member
PLEASE please read sir . I am dedicating 10 hours of my time in writing every line or questions that u see . Yes it is a stress for u to see my long paragraphs as u have other works to do . I dont know what else can i do if i dont get clear of my doubts other than leaving mathematics permanently. U guys are only hope . No one is there to help me .

we can call 4k and 2k
I actually wanted a reply to my " Specific questions" as i had these doubts naturally within the last 4 days and i am now stressed really . If u have time, please go through it once again the OP.

Yes i did the same thing when i was solving on my own . That 'K' that u took is HCF actually. I did it like this -> When i saw the question i knew from the ratio 4:2 that it has to be in simpler form (as Rs 4 + Rs 2 does not add literally up to Rs12. ) i.e 4:2 ratio is given after reducing x : y where X represents the actual share/Amount (the money out of 12) of first person and y (the remaining money out of 12) of the second person

So I know that 4:2 has been reduced from x:y with the use of hcf of x,y.
So 4* hcf : 2* hcf = x : y

also i know that x + y adds up to rs 12 so 4 hcf +2hcf =6hcf => hcf = 12/6 = rs '2'
THIS method was much satisfying to me coz this was my logic . After doing this method i came to a conclusion that 4:2 this ratio actually represented a simpler form of x:y (the ratio actually represented a simpler version of each "share size" rs 8 : rs 4 )

But when i saw the actual solution that i had given in the OP I was confused when i saw this line
we need to take 4 of rs2 and 2 of rs2
I thought the ratio 4:2 was actually representing
ratio between " How many GROUPS of each part value taken by the first person : second "
Q2: What does 4 : 2 actually represent?
I am still confused about "What does 4 : 2 actually represent?" . WHAT i feel is that it represents both the things that i quoted in my Op doubt 2 .

Q1: Why did they add 4 +2?
Q3: WHY DO WE DIVIDE 12 by 6?
U replied to this above two questions that
We added 4+2 because we wanted a sum.
We divide by 6 because we want to find k.
See when they are adding directly 4+2 without doing it like this " 4k+2k" thats where the confusion arose to me.
Why they omitted the 'k or hcf' part .

You have 12 of them. To make a ratio of 4:2, you can just put 4 into one pile and 2 into another. That uses up Rs 6. You have another 6, so you repeat; now you have Rs 8 in one pile and Rs 4 in the other.
ok. But actually i was not wanting the concrete eg.

Observe that adding 4+2 arises naturally in keeping track of how much is used up in each round
There are two rounds . ok

although we didn't actually divide by 12, we could have done so to find how many rounds we needed.
I am not sure what u said here.
As far i understood that i know i have 12 rs in total so if i do in the piling method way (taking 4 of one rupee coins in 1 pile and 2 of rs 2 coin in another pile per round) ; in 1 round i will have Rs 6 .

So creating a quotative division expression --> rs 12 / rs 6 per round = 2 rounds

If this is what you are trying to mean then why did u said "we didn't actually divide by 12" I am not dividing by 12 but dividing 12 by 6 to get 2 rounds.

THANKS A LOT

#### Dr.Peterson

##### Elite Member
PLEASE please read sir . I am dedicating 10 hours of my time in writing every line or questions that u see . Yes it is a stress for u to see my long paragraphs as u have other works to do . I dont know what else can i do if i dont get clear of my doubts other than leaving mathematics permanently. U guys are only hope . No one is there to help me .
I am recommending that you "leave mathematics" in the form you are pursuing, because it is not mathematics. Talk of partitive and quotative problems is not mathematics; it is used in teaching (or rather, in teaching teachers of) elementary mathematics, in that it is only about different ways in which division can be applied in concrete situations. That is important for initial learners, but it is not mathematics proper, because mathematics is abstract.

What you are doing is trying to analyze a particular method of explaining the solution of this problem, which is not needed. That's why I showed you a couple other methods, one purely abstract and the other using a different concrete model. (And, yes, I mistyped the statement about dividing by 12, when I meant "dividing 12 by 6".)

Dropping your attempt to analyze everything in these terms will remove the stress, and return you to actual mathematics.

But let's look at what you say about the particular explanation you are asking about, since it troubles you so much. (I do wish, however, that you would show exactly the actual work you are asking about, rather than quoting it indirectly and out of context.)

But when i saw the actual solution that i had given in the OP I was confused when i saw this line

we need to take 4 of rs2 and 2 of rs2​

I thought the ratio 4:2 was actually representing

ratio between " How many GROUPS of each part value taken by the first person : second "​

Q2: What does 4 : 2 actually represent?​

I am still confused about "What does 4 : 2 actually represent?" . WHAT i feel is that it represents both the things that i quoted in my Op doubt 2 .
The problem they are solving is:

DIVIDE Rs 12 into 4:2 between 2 persons​

In the problem, 4:2 is simply a ratio. In itself, it is abstract; but it is being applied as the ratio of two quantities of money, which you can think of as parts of the 12. We have 12 rupees, RRRRRRRRRRRR, and are to divide them into two parts, which turn out to be RRRRRRRR RRRR, to give to two people. So, yes, we want 4:2 to be equivalent to X:Y (that is, 8:4).

I can't make much grammatical sense of your proposals, but it is true that a ratio can be seen in multiple ways (that, in a sense, is my whole point -- you don't have to pick one view as "what it really is"!).

Q1: Why did they add 4 +2?
Q3: WHY DO WE DIVIDE 12 by 6?

U replied to this above two questions that

We added 4+2 because we wanted a sum.​
We divide by 6 because we want to find k.​
To the extent that your questions were about the specific method you were discussing, I was not answering them; I was talking about what I did in my abstract methods, to show that your questions don't matter at all.

See when they are adding directly 4+2 without doing it like this " 4k+2k" thats where the confusion arose to me.
Why they omitted the 'k or hcf' part .
The explanation you are asking about is concrete, in effect! That's what you are asking about: groups and parts and persons. My point is that you can solve it abstractly, which does away with those questions. This is the essence of mathematics: To change "add 3 more sheep to my 2 sheep" into merely "2+3". (My "k" and your "hcf" are part of the abstract approach, so I'm glad you wish for them.)

Anyway, the explanation you are asking about looks at the ratio 4:2 and says, we want our 12 rupees to be split into equal "parts": 4 parts to me and 2 parts to you. The goal is to figure out how many rupees are in each "part". (This notion of "parts" is traditional in talking about ratios, and is not part of the "quotative and partitive" idea.)

There are a total of 6 "parts": _ _ _ _ _ _. That's why we add 4+2.

To split 12 rupees into 6 equal parts, each part must consist of 2 rupees. That's why we divide by 2.

So I get 4 parts, each of which is 2 rupees: RR RR RR RR, for a total of 8 rupees.

You get 2 part, each of which is 2 rupees: RR RR, for a total of 4 rupees.

Again, here is the explanation you quoted or paraphrased:

"They are adding the ratio terms 4 and 2 getting 6 by which 12 will be divided​
Rs12/6parts=2Rs/ part .​
Now they say we need to take 4 of rs2 and 2 of rs2​
So the shares are rs 8 and rs 4 "​

No, that's not a quote, is it? If you quoted exactly what they said, I could probably help you understand their language; but it probably means essentially what I just explained: 4:2 means 6 parts, and 12/6 = 2 so each part must be 2 rupees. Therefore the two people get 4 and 2 times 2 rupees respectively, making 8 and 4.

pka

#### Saumyojit

##### Full Member
I mistyped the statement about dividing by 12
U would not belive that when you said divide by 12 i thought you were right .so i spent 3 hours in figuring out what you try to mean and then i got the feeling you actually mistyped. I also mistyped
2 of rs 2 coin in another pile per round
But thanks a lot in replying.
which you can think of as parts of the 12
What thing will i think of as parts of 12 ; are u talking about "Rs 8 and Rs 4 " which are not the parts of 12 but are the respective sizes of share of each one part out of 2 unequal parts.
This is the essence of mathematics: To change "add 3 more sheep to my 2 sheep" into merely "2+3"
i did not know that there is a concrete approach and abstract approach till date. The abstract is far better and intutive . Ok thanks

we want our 12 rupees to be split into equal "parts":
I dont understand why rs 12 has to be splitted into equal parts. Yes i went through ur post several times .
If the question was given like this " divide rs 12 between 2 persons equally " then i should be dividing 12 by 2 to get 2 equal parts of each rs 6 but they have clearly said the share will not be equal then why i am dividing 12/6 .
12 /6 can also be done if it was said there are 6 persons for rs 12 and so how much each person will take .This satisfies my logic of dividing 12 /6

What i feel that there is "4 parts of somevalue and 2 parts of another some value " = 6 parts of Rs12

This is another issue that arose in my mind just now . --> i thought " 4:2 can it mean that 12 has been divided into 4 parts (rs 12 / 4 parts = rs 3 per part) and has been divided into 2 parts ( rs 12/2 parts = rs 6 per part) " I tried to think why it cannot happen and got the reason which is unsatisfying to me that rs 12 is present two times i.e rs 24 is present which is violating the question.

I also know this statement which arose in my mind some how while writing cannot happen but why ?

(This notion of "parts" is traditional in talking about ratios, and is not part of the "quotative and partitive" idea.)
What do u mean and to which part of my comment are u giving this reply .

#### Dr.Peterson

##### Elite Member
Please quote the actual solution you are talking about, so we can discuss that. We need a specific context. I've asked for this several times.

In fact, you said "most of the solutions" say the same thing; is that really true?

Here is what I meant in saying that "parts" is a traditional term in relation to ratios: We commonly describe, say, a mixture in the ratio of 1:3 as "one part this to three parts that". That is, whatever unit we use, there is one unit of one and three units of the other. It might be 1 gallon to 3 gallons, or 1 kg to 3 kg, or 1 unit of 2.5 liters to 3 units of 2.5 liters, or whatever. We use the word "part" to refer to that common unit; what is essential is that the "parts" are all the same size.

So if we want to divide money in the ratio 4:2, it means one person get 4 "somethings" when the other gets 2 of the same "somethings". This is the same idea as the abstract "4k to 2k", where k may be any number.

So our ratio looks like this, where each box must contain the same amount of money:

We want to distribute 12 rupees equally into those boxes:

How do we do that? There are a total of 6 boxes to fit 12 things, so we put two rupees in each box:

So one person gets 8 and the other gets 4. It's that simple.

That's the concrete approach on which the explanation is based. When you grow up, you don't need that. You can just say, I need 4 "parts" for one person and 2 for the other; that makes a total of 6 "parts", and since 12 is 6 times 2, each part consists of 2. So the 4 parts are 8, and the 2 parts are 4.

When you really grow up, you don't need words, but can use symbols: The ratio 4:2 means the two amounts are 4k and 2k for some quantity k; we need 4k+2k = 12, so 6k = 12, and k = 12/6 = 2. Therefore the amounts are 4*2=8 and 4*2=4.

This should eliminate any need for your questions. But let's try to answer them:

What thing will i think of as parts of 12 ; are u talking about "Rs 8 and Rs 4 " which are not the parts of 12 but are the respective sizes of share of each one part out of 2 unequal parts.
I used the word "parts" in different ways at different times, because that's how language works. I think you're referring to my initial statement of the problem, as splitting the 12 rupees into two unequal "parts" or "portions". Yes, of course I was referring then to the 8 and 4. If I had 12 blocks and put them into piles of 8 and 4, isn't it valid to call those "parts" of the 12? You seem to be trying to turn language into a formal system where every word has exactly one meaning, which it isn't. No wonder you have trouble understanding anyone.

I dont understand why rs 12 has to be splitted into equal parts. Yes i went through ur post several times .
If the question was given like this " divide rs 12 between 2 persons equally " then i should be dividing 12 by 2 to get 2 equal parts of each rs 6 but they have clearly said the share will not be equal then why i am dividing 12/6 .
12 /6 can also be done if it was said there are 6 persons for rs 12 and so how much each person will take .This satisfies my logic of dividing 12 /6

What i feel that there is "4 parts of some value and 2 parts of another some value " = 6 parts of Rs12
Yes, 4 parts and 2 parts of the same size. And that size, 2, is found by dividing 12 by 6. Do you understand yet?

This is another issue that arose in my mind just now . --> i thought " 4:2 can it mean that 12 has been divided into 4 parts (rs 12 / 4 parts = rs 3 per part) and has been divided into 2 parts ( rs 12/2 parts = rs 6 per part) " I tried to think why it cannot happen and got the reason which is unsatisfying to me that rs 12 is present two times i.e rs 24 is present which is violating the question.

I also know this statement which arose in my mind some how while writing cannot happen but why ?
Do you understand yet why we don't divide 12 by 4 here? None of what you say here makes any sense. We divide by 6 because that is the total number of equal parts, into which the total number 12 has to be divided.

#### Saumyojit

##### Full Member
"parts" is a traditional term in relation to ratios: We commonly describe, say, a mixture in the ratio of 1:3 as "one part this to three parts that"
Yes i saw from the internet that " part" means a part of the whole and another meaning was " a mix of one part cement to five parts ballast" that means i may have 1 gm of cement for every 5 gms of ballast ..that may not literally mean what actual amount of each thing i have . I may have 5gm of cement for every 25 gms of ballast in the mixture.

This notion of "parts" is traditional in talking about ratios, and is not part of the "quotative and partitive" idea.
Okay i got what u meant ; u are trying to tell me that dont take "part" as the meaning like if something x is divided into y parts then we do x/y . ??

But i want to know why did u said this line . What did u saw in my which comment of which no post that u thought i was confusing the meaning of parts.

We want to distribute 12 rupees equally into those boxes
MAIN DOUBT:
UNDERSTAND one thing -> If the question was given like this "Divide rs 12 among 6 persons/parts/boxes " then its straightforward for me to understand why we divide rs 12/6 but in this particular question when its not direct ;

as far as i understood this type of trick question is telling that " the ratio terms 4 and 2 advises the answerer to think as the first person is taking 4 of somethings and the second is taking 2 of somethings out of whole rs 12 ; that somethings inherently translates to "each part value of rs 12" ; so to find the size of each part i need to know how many parts are there ; so i add .....as 4 of something + 2 of something gives me 6 of everything i.e 12 therefore divide 12 by 6 to get value of something as rs 2 .
4 parts and 2 parts of the same size
I did not knew that this above statement was valid or not . I thought 4 groups and 2 groups of the same size or 4 of rs 2 and 2 of rs 2 or 4 groups of rs 2 and 2 groups of rs 2 are valid till now but not "4 parts of rs 2" as for me part meant only part of a whole but i thought writing 4 parts of rs 8 is valid .

but u said
You seem to be trying to turn language into a formal system where every word has exactly one meaning, which it isn't
does this mean writing "4 parts and 2 parts of the same size" and "calling rs 8 and rs 4 as the parts of 12" are both valid then?

#### Dr.Peterson

##### Elite Member
Yes i saw from the internet that " part" means a part of the whole and another meaning was " a mix of one part cement to five parts ballast" that means i may have 1 gm of cement for every 5 gms of ballast ..that may not literally mean what actual amount of each thing i have . I may have 5gm of cement for every 25 gms of ballast in the mixture.
This is the same idea as equivalent ratios. 1:5 is the same as 5:25.

Okay i got what u meant ; u are trying to tell me that dont take "part" as the meaning like if something x is divided into y parts then we do x/y . ??

But i want to know why did u said this line . What did u saw in my which comment of which no post that u thought i was confusing the meaning of parts.
I wasn't directly replying to anything specific that you said, only reminding you of the general fact that words mean different things in different contexts. You have (in this thread and elsewhere) mentioned partitive and quotative, and you are taking words like "part" too rigidly, and I don't want you to. If you don't think you are doing that, then ignore the comment.

MAIN DOUBT:
UNDERSTAND one thing -> If the question was given like this "Divide rs 12 among 6 persons/parts/boxes " then its straightforward for me to understand why we divide rs 12/6 but in this particular question when its not direct ;

as far as i understood this type of trick question is telling that " the ratio terms 4 and 2 advises the answerer to think as the first person is taking 4 of somethings and the second is taking 2 of somethings out of whole rs 12 ; that somethings inherently translates to "each part value of rs 12" ; so to find the size of each part i need to know how many parts are there ; so i add .....as 4 of something + 2 of something gives me 6 of everything i.e 12 therefore divide 12 by 6 to get value of something as rs 2 .
It is not a "trick question". But I think you are saying the right thing here. If you think it is too tricky, then use a different method!

I did not knew that this above statement was valid or not . I thought 4 groups and 2 groups of the same size or 4 of rs 2 and 2 of rs 2 or 4 groups of rs 2 and 2 groups of rs 2 are valid till now but not "4 parts of rs 2" as for me part meant only part of a whole but i thought writing 4 parts of rs 8 is valid .
Here you again seem to be taking the word "part" too rigidly! The word "part" just means "part"; whether all "parts" are the same size is a matter of context. In the context of talking about 4:2 as 4 parts to 2 parts, they are understood to be equal parts; if you just give me part of a cake, it can be anything. This is the way language works.

does this mean writing "4 parts and 2 parts of the same size" and "calling rs 8 and rs 4 as the parts of 12" are both valid then?
Yes. "Parts is parts" and nothing more.

#### Saumyojit

##### Full Member
Here you again seem to be taking the word "part" too rigidly! .
Yes thats because i thought part meant a fragment of a whole. Thats why i thought 4 parts of rs 8 (rs 8 is value of whole or value of all total 4 parts ) not 4 parts of rs 2 ( as rs 2 is not the value of all total 4 parts but single part value ) .

I can make one expression by asking Rs'12' has been divided into how many parts so that one of the part is rs '8'
12/X=8
X=3/2
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
Can i interpret the same problemn in this way ? The only difference is that the ratio then would become 1 : 0.5
but the amount recieving is same .

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#### Dr.Peterson

##### Elite Member
Yes thats because i thought part meant a fragment of a whole. Thats why i thought 4 parts of rs 8 (whole) not 4 parts of rs 2 ( as rs 2 not whole but single part value) .
Suddenly it appears that what you are confused by is not the word "part" but the word "of". Prepositions are very subtle in any language, so that's not too surprising. But this is a site about math, not about teaching English, which I suspect is a big part of what you need.

If someone says, "4 parts of rs 8" as you say above, the "of" would mean what each is taken from, that is, 8 is the whole. If they say "4 parts of rs 2", the "of" would mean "consisting of", making 2 the size of each part. These look similar, and you would have to determine which is meant by context, which I have mentioned many times.

I don't see that I've use either form; but this appears to be (at least part of) what you are bothered by in what you read somewhere else. I had assumed this was your own wording, and therefore you knew what it meant:
Now they say we need to take 4 of rs2 and 2 of rs2
Is this is a direct quote from your source, though not presented as such? (I wish you would quote it as I have asked repeatedly, so I could be sure what is your own wording and what is theirs.)

I would not say it exactly that way; it is not natural English to me. But what they mean is clearly 4 parts (each of which is Rs 2) and 2 parts (each of which is Rs 2). The "of" here means "consisting of"; I might say "I want 4 of these and 2 of those".

#### Saumyojit

##### Full Member
Suddenly it appears that what you are confused by is not the word "part" but the word "of"
Thats why i said
What did u saw in my which comment of which no post that u thought i was confusing the meaning of parts.
I was confusing the using of the word "part" with the word "of"

{Parts mean Division , don't think because I wrote parts of 12 that means 'of' means multiplication so I multiply 6*12 ;! no that is wrong ..
I can write 1/6 of 12 or 6 parts of 12
Groups & parts are different}
I hope u read this part in my OP
"4 parts of rs 8" as you say above, the "of" would mean what each is taken from
exactly this is what i was saying but
"4 parts of rs 2", the "of" would mean "consisting of", making 2 the size of each part. These look similar, and you would have to determine which is meant by context
Acc to my knowledge 4 parts of rs 2 meant till now as rs 2 has to be divided into 4 parts .

4 of rs '2' this was exactly quoted from the solution and i knew that 4 groups of rs 2 sounds right but not "4 parts of rs 2" .
When i am saying " 4 groups of rs 2" or " 4 of rs 2" I would mean "consisting of" or repetition of 2 four times .

BUT when i am using the word "part" with "of" then i know i have to write 4 parts of whole something i.e rs 8 not rs'2' as for me part meant "fragment of a whole" but this is what i did not realize till now
"4 parts of rs 2", the "of" would mean "consisting of"
That i could use "parts" with "of " and at the same time it can mean " consisting of"
I always thought using "parts" with "of" will only mean "what each is taken from "

4 parts (each of which is Rs 2) and 2 parts (each of which is Rs 2). The "of" here means "consisting of"; I might say "I want 4 of these and 2 of those".
YES.

Can i interpret the same problemn in this way ? The only difference is that the ratio then would become 1 : 0.5
but the amount recieving is same .

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#### Dr.Peterson

##### Elite Member
There are too many quotes of quotes of quotes out of context; I can't follow what you are asking when you do that.

If you really still have questions, please do two things:
1. QUOTE THE ACTUAL PAGES YOU WERE ORIGINALLY OBJECTING TO, WHICH YOU PARAPHASED, AS I HAVE ASKED OVER AND OVER. I WILL NOT RESPOND AGAIN WITHOUT THIS.
2. Ask a complete question without reference to past statements, so I can see what you are asking all in one place.

#### JeffM

##### Elite Member
I commend Dr. Peterson for all the time and patience he has.

You said in one post, that it is a “trick” question, and Dr. Peterson said that he did not get why the ratio described as 4:2 was not described as 2:1. To my mind, Dr. Peterson has identified the “trick” in this question: the ratio has not been expressed in lowest terms, which would perhaps have made explaining the process more intuitive.

Let’s go back to the original problem and ask what it means. It is asking you to divide the quantity of 12 into two UNEQUAL parts.

How do I know that the number of parts is 2? Because the ratio has two terms.

How do I know that the parts are not equal? Because the ratio is not 1:1.

Direct division divides into equal parts so we will not get a correct answer dividing 12 by 2. If we do that we will indeed get 2 parts, each of 6 pieces, which is an equal number of pieces.

Now you can think about it this way. For every 4 pieces that person A has in A's part, we want person B to have 2 pieces in B's part. That requires 6 pieces in total. If I divide 6 into 12, I get 6 tiny parts of 2 pieces each. So if I give 4 of those tiny but equal parts to person A and 2 of those tiny but equal parts to person B, person A's share will be in the ratio of 4 pieces to every 2 in B's share. And each of the tiny but equal parts contains 2 pieces so A will have a share consisting of 4 times 2 or 8 pieces. B gets 2 of those tiny but equal parts each containing so B's share will contain 2 times 2 or 4 pieces. 8 + 4 = 12. And 8 is to 4 as 4 is to 2, namely double.

So our general procedure is

(1) Add up the terms of the ratio. In this case that sum was 6.
(2) Divide the total by that sum so get the number of pieces in tiny but equal parts. In this case, that quotient is 2.
(3) Create the shares by assigning the number of tiny but equal parts in accordance with the terms of the ratio. In this case, that is 4 tiny parts to A, which is equal to 8 pieces, and 2 tiny parts to B, which is equal to 4 pieces.
(4) Check that the number of pieces in each share are in the proper ratio and that the sum of the pieces in all the shares adds up to the total..