Unitary & Division

[Saumyojit]

1St Doubt: The question is 6 apples divided among 7 persons

Case1&2 are just vice versa so as 3&4

Two different interpretations in Unitary Method
Case1: 6 apples divided among ->7 People

So 1 apple eaten by -->7/6 people or Person(hypothetical situation)

Or if I look it from another way
Case 2
7 People will get total->6 apples
1 person will get -->6/7 of a apple ( which is possible)

IF THE question is 6 apples divided among 12 persons
Case 3:
6 apples divided among ->12 persons
1 apple shall be eaten by -->2 person
Case 4:
12 persons will get ->6 apples
1 person will get -->1/2 of apple

is case 1 invalid ?if it is not then 7/6 people does not make any sense
Then if i have to make case 1 possible then i have to take both divisor and dividend Exact multiple of each other just like in case 3
Right?


2nd Doubt:

suppose the question is what is 30/100 of 200 then When I am reading that what is "30 percent of 200" i see that 'of ' is there(of means multiplication), Then I directly multiply (30 * 200 )/100
DOUBT: The whole thing is translating into
30/100/200

From the Wikipedia page:

"If there are multiple divisions in a row
a/b/c translates to either

Case 1:a/b/c=(a/b)/c=a/(b*c) [Left associative which is default for divison]
Or
Case 2:a/b/c=a/(b/c)=(a*c)/b " [Right associative]

30/100/200 this is actually taking 2nd case form while executing

30/(100/200)=(30*200)/100

BUT watch carefully in 2nd case when there is paranthesis around b/c then b/c is a single term & a is another term

CONTRADICTION: but in 30/100/200 ; 30/100 is a single term and 200 is another term as the question is 30/100 of 200

according to that it should follow the first case while executing. i.e a/(b*c) = 30/(100*200)=30/20000
@Dr.Peterson @HallsofIvy @JeffM

 

[JeffM]

You are mixing up different things.

You ask a question that arises in abstract algebra where the associative property is a property of binary operations. Let @ represent ANY binary operator. Thus, at a very high level of technical formality, you should never even write a @ b @ c. It is meaningless because the operator is only defined in terms of two operands, and you have specified three. We may however write (a @ b) @ c or a @ (b @ c). Are they always equal to each other? If yes, @ represents an associative operation. If no, @ does not represent an associative operation. It is that simple; it is a definition. And PEMDAS is not a rule in abstract algebra; it can’t be because we have no clue what the operations are except as they are described by properties.

You then jump to elementary arithmetic, presumably supplemented by PEMDAS, interpreted in physical terms.

You say there is no such thing physically as 7/6 of a human being. So what? You admit that there is a physical answer, namely each of six of the human beings receives less than a whole apple while the seventh receives six or more small pieces from different apples. And you almost certainly cannot exactly cut six different apples of different shapes and masses into six sevenths that are equal. In practical terms, 6/7 of an apple is just as fictional as 7/6 of a person. I have said now several times that your persistence in terms of thinking in analogies is a hindrance to understanding mathematics. Mathematics helps us understand the real world by simplifying the real world into a simple but fictional world. If mathematics says that two answers are exactly equivalent in the fictional world, and at least one answer can be approximated in the physical world, mathematics has been useful.

And finally you start playing word games. ”Right associative” is a term you made up. It has no mathematical meaning. And 30% of 200 does not translate

[MATH]30 \div 100 \div 200.[/MATH]
Obviously if you do not yet know how to do percentage problems in basic arithmetic, you are not ready to study abstract algebra.

 

[HallsofIvy]

1St Doubt: The question is 6 apples divided among 7 persons

Case1&2 are just vice versa so as 3&4

Two different interpretations in Unitary Method
Case1: 6 apples divided among ->7 People

So 1 apple eaten by -->7/6 people or Person(hypothetical situation)

WHY would you write that? Are you writing "7/6" just because the "7" comes before the "6" in the sentence?? I hope you understand that "grammar" is NOT "mathematics!

Also, did you not notice that 7/6 is larger than one? If 6 apples are divided amoung 7 people, each person does NOT get more than one apple!

Or if I look it from another way
Case 2
7 People will get total->6 apples
1 person will get -->6/7 of a apple ( which is possible)

Yes, mathematics is not just applying formulas blindy. It requires thinking!

IF THE question is 6 apples divided among 12 persons

Case 3:
6 apples divided among ->12 persons
1 apple shall be eaten by -->2 person

So each person gets 1/2 apple.

Case 4:
12 persons will get ->6 apples
1 person will get -->1/2 of apple

is case 1 invalid ?if it is not then 7/6 people does not make any sense

Yes, "case 1" in the first problem, is invalid. In fact that calculation does not make sense..

But your "case 1" in the second problem is NOT the same as "case 1" in the second problem. In the second problem, the two "cases" give the same answer, 1/2 apple per person.
Then if i have to make case 1 possible then i have to take both divisor and dividend Exact multiple of each other just like in case 3
Right?

2nd Doubt:

suppose the question is what is 30/100 of 200 then When I am reading that what is "30 percent of 200" i see that 'of ' is there(of means multiplication), Then I directly multiply (30 * 200 )/100

Yes, that is correct. And you can calculate that as (30)(200/100)= 30(2)= 60 or as (30/100)(200)= 0.3(200)=60.

DOUBT: The whole thing is translating into
30/100/200

What??? No,the 200 is multiplying 30/100, not dividing it!

From the Wikipedia page:

"If there are multiple divisions in a row
a/b/c translates to either

Case 1:a/b/c=(a/b)/c=a/(b*c) [Left associative which is default for divison]
Or
Case 2:a/b/c=a/(b/c)=(a*c)/b " [Right associative]

30/100/200 this is actually taking 2nd case form while executing.

Yes, but (30/100)(200) is NOT 30/100/200! The 200 multiplies, it does not divide.

30/(100/200)=(30*200)/100

BUT watch carefully in 2nd case when there is paranthesis around b/c then b/c is a single term & a is another term

Yes, and that parenthesis is what makes this different from what you did before.
What Wikipedia was trying to tell you is that 30/100/200 is ambiguous and should NOT be used..

CONTRADICTION: but in 30/100/200 ; 30/100 is a single term and 200 is another term as the question is 30/100 of 200

No, "30/100/200" is nonsense and doesn't mean anything!

according to that it should follow the first case while executing. i.e a/(b*c) = 30/(100*200)=30/20000
@Dr.Peterson @HallsofIvy @JeffM

 

[Saumyojit]

WHY would you write that? Are you writing "7/6" just because the "7" comes before the "6" in the sentence?? I hope you understand that "grammar" is NOT "mathematics!

@HallsofIvy See in unitary method information /relation is given between two things.

if 7 people all total gets 6 apples
then 1 person will get 6/7 th of a apple
So same way bringing "apples" on the lhs and "people" on the right side

6 apple have to be divided among ->7 People
so 1 apple will be divided among 7/6 people

I did this thing by logic not by because the "7" comes before the "6" in the sentence . And If case 3 makes sense (finding in terms of 1 apple) why would u say that "case 1" in the first problem, is invalid. In fact that calculation does not make sense "

WHICH IS why i made up another eg right now .
NEW EG:
If 12 apples is divide among 6 people ( apple quantity is more than person)
Same way in case 1 and case 3 (KEEPING no of apples in left side)

12 apple have to be divided among ->6 People
so 1 apple will be divided among 1/2 of a person which means 1 person will get 2 apple . SEE here 1/2 of a person does not makes sense which is why jeffm said

there is no such thing physically as 7/6 of a human being. So what? You admit that there is a physical answer. Mathematics helps us understand the real world by simplifying the real world into a simple but fictional world !

that fictional world is 1/2 of a person or 7/6 people but if 1/2 of a person gets 1 apple then 1 each person will get 2 (directly proportional) thats how from mathemtics (fricitional world) we can make sense.

did you not notice that 7/6 is larger than one? If 6 apples are divided among 7 people, each person does NOT get more than one apple!

EACH person gets 6 slices which is not whole . NO body gets a whole so there is no question of getting more than one apple .

But your "case 1" in the second problem is NOT the same as "case 1" in the second problem. In the second problem, the two "cases" give the same answer, 1/2 apple per person.

if u see case 1 and case 2 also (of 1st problemn) two cases will give same answer each person gets 6/7 of a apple .

But One thing i would like to compare between CASE 1 and the new eg above ; in case 1 one apple gets eaten by 7/6 people and in the new eg 1 apple is eaten by 1/2 of person . ALTHOUGH BOTH ARE FICTIONAL. i dont know why in the new eg case its easier to visualize that if 1/2 person gets 1 apple then 1 person will get 2 apples (May be coz i can INSTANTLY double 1/2 and 1 gives me 1 person and 2 apple) but in the 1st CASE its hard to visualize of 7/6 although i know the technique that if 7/6 people gets 1 apple ; so to convert it into 1 person i need to multiply 7/6 with 6/7 and also 1 apple with 6/7 . (directly proportional)

namely each of six of the human beings receives less than a whole apple while the seventh receives six or more small pieces from different apples. And you almost certainly cannot exactly cut six different apples of different shapes and masses into six sevenths that are equal.

@JeffM I absolutely agree the seventh recieve six slices from diferent apples but why do u say "more small pieces" and "different apples of different shapes" . See Everybody' s share is : 6/7 +6/7+6/7+6/7+6/7+6/7+6/7 which gives us 7 apples originally with which i STARTED.

WE had a discussuion long ago (u, me ,dr peterson) in my post "FRACTIONS" where u told we add fractions of the same unit(i.e 7 th in this case) and the individual objects have to be of the same size OR it cannot happen that I took half of one pizza and u took another half of a bigger pizza and we ate 1 whole pizza together.
SO WHY APPLES ARE OF DIFFERENT SIZE? they have to be same size

 

[Saumyojit]

@Dr.Peterson say something in this post and also reply to my post no 4 in this question

 

[HallsofIvy]

@HallsofIvy See in unitary method information /relation is given between two things.

if 7 people all total gets 6 apples
then 1 person will get 6/7 th of a apple
So same way bringing "apples" on the lhs and "people" on the right side

6 apple have to be divided among ->7 People
so 1 apple will be divided among 7/6 people

I did this thing by logic not by because the "7" comes before the "6" in the sentence . And If case 3 makes sense (finding in terms of 1 apple) why would u say that "case 1" in the first problem, is invalid. In fact that calculation does not make sense "

WHICH IS why i made up another eg right now .
NEW EG:
If 12 apples is divide among 6 people ( apple quantity is more than person)
Same way in case 1 and case 3 (KEEPING no of apples in left side)

12 apple have to be divided among ->6 People
so 1 apple will be divided among 1/2 of a person which means 1 person will get 2 apple . SEE here 1/2 of a person does not makes sense which is why jeffm said
that fictional world is 1/2 of a person or 7/6 people but if 1/2 of a person gets 1 apple then 1 each person will get 2 (directly proportional) thats how from mathemtics (fricitional world) we can make sense.

EACH person gets 6 slices which is not whole . NO body gets a whole so there is no question of getting more than one apple .


if u see case 1 and case 2 also (of 1st problemn) two cases will give same answer each person gets 6/7 of a apple .

But One thing i would like to compare between CASE 1 and the new eg above ; in case 1 one apple gets eaten by 7/6 people and in the new eg 1 apple is eaten by 1/2 of person . ALTHOUGH BOTH ARE FICTIONAL. i dont know why in the new eg case its easier to visualize that if 1/2 person gets 1 apple then 1 person will get 2 apples (May be coz i can INSTANTLY double 1/2 and 1 gives me 1 person and 2 apple) but in the 1st CASE its hard to visualize of 7/6 although i know the technique that if 7/6 people gets 1 apple ; so to convert it into 1 person i need to multiply 7/6 with 6/7 and also 1 apple with 6/7 . (directly proportional)
@JeffM I absolutely agree the seventh recieve six slices from diferent apples but why do u say "more small pieces" and "different apples of different shapes" . See Everybody' s share is : 6/7 +6/7+6/7+6/7+6/7+6/7+6/7 which gives us 7 apples originally with which i STARTED.

WE had a discussuion long ago (u, me ,dr peterson) in my post "FRACTIONS" where u told we add fractions of the same unit(i.e 7 th in this case) and the individual objects have to be of the same size OR it cannot happen that I took half of one pizza and u took another half of a bigger pizza and we ate 1 whole pizza together.
SO WHY APPLES ARE OF DIFFERENT SIZE? they have to be same size

Have you told the apples that?

 

[Saumyojit]

@Harry_the_cat @Jomo please see the post

 

[Saumyojit]

Have you told the apples that?

to which part are u referring

the individual objects have to be of the same size OR it cannot happen that I took half of one pizza and u took another half of a bigger pizza and we ate 1 whole pizza together.
SO WHY APPLES ARE OF DIFFERENT SIZE? they have to be same size

this last part ..
i wrote so many things in post 4 referring to both @HallsofIvy and @JeffM previous comments and its been 3 days since i did NOT get a elaborate breakdown reply.
please help me i beg of u ..
i just want reply to each of my doubt in POST 4 especially the last part to @JeffM comment

 

[Harry_the_cat]

@Harry_the_cat @Jomo please see the post

Why me?

 

[Saumyojit]

Why me?

Because nobody is replying since hall of ivy last post . I am stiil wiaitng

 

[Saumyojit]

@Dr.Peterson please share ur views on post #4
@JeffM @tkhunny

 

[Dr.Peterson]

@Dr.Peterson please share ur views on post #4
@JeffM @tkhunny

I've chosen to stay out of this thread, because it is too complicated to follow. If you want to ask a question about it in a new thread, I may choose to answer if it is clear enough what you are asking.

 

[JeffM]

I'll only try to respond to my part of post 4.

If we had six apples that were perfectly equal in weight, flavor, nutrition, etc.

then one way to proceed is to divide each apple into perfectly equal sevenths, leading to 42 pieces each equal to a seventh of an apple, and distribute 6 pieces to each person.

Another way to go is to is to remove exactly one seventh from each apple, leading to 6 pieces each equal to a seventh of an apple and 6 cut apples each equal to 6/7ths of a full apple. We could give six people a cut apple each and the seventh person the six pieces.

Both procedures lead to the same result: each person gets 6/7ths of an apple.

My point there is that if we insist on thinking in terms of physical processes, there are frequently different ways to go about it. It's a minor point.

My other point was about the ideal aspect of mathematics and why thinking physically is frequently not helpful. Try going to market and actually buying 6 apples that are regular enough in shape to cut into perfect sevenths and that are exactly equal in weight, flavor, nutrition, etc. In the real world, this problem can only be solved approximately.

It's really not worth worrying about my comment.