proportion with more than 2 variables

Saumyojit

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I have questions in every step. Please help as soon as possible. I am in very much anxiety understanding this. Please explain in simple words. My brain is slow.


I was reading a article where it is said "if a is proportional to b then if any value of a,suppose a0 becomes 2a0 then b0 will change the same way as a0 i.e become 2b0

2a0/2b0=a0/b0=Constant "

if a is 2 and b is 3 then 2a=4 and b will be 2b =6. ok got it .

MY problemns starts from here :

after that it is written : "one may have a situation in which more than two variables are proportional to each other. For instance, we might have a situation in which a is proportional both to b and to c . In these situations, the above procedure only works if we keep values other than the two under considerations constant. More generally, using the exact same method as above, we can combine the proportionalities:

a/(b×c)= Constant (Since a is proportional to both b and c so all three variables change “together” ) "

after reading this part my understanding is this :if a is 2 ,b is 3 , c is 4 as they have told a is proportional to both b and c so values of b and c will change “together” in the same way as a changes. I am saying this line because in the direct proportion they have said that "if a is proportional to b then b will change the same way as a changes" . SO in this case of 3 variables i am applying the same logic of a ∝ b just here change is a ∝ bc so both the value of b and c will change depending on how a is changing .



doubt 1: if i double a to 2a =4 ; b will become 2b=6 ; c will become 2c=8 as (a is proportional to both b and c so b and c will change as a changes)
then again if i double 2a to 2 *2a =8, 2b to 2*2b=12, 2*2c=16

but if i do it like this a/(b*c) =k does not hold

they have said all three variables will change together. Am i right? HOW does the changing when there is more than 2 variables. Then arises the next question

doubt 2 : What do they mean by saying "we keep values other than the two under considerations constant" ? and why

Then they have said "Note that this equality captures automatically the fact that if a∝b and a∝c then b and c are inversely proportional to each other. The reason is that by “inversely proportional” mean precisely that if b changes a certain way, then c changes in such a way that 1/c (i.e. its inverse) changes in the exact same way. Thus, we can express the inverse proportionality between b and c as

b×c= Constant "

NOTE THAT my thinking process is like this : When i read up a∝b and a∝c just like i said before I assured myself thinking of the logic of direct proportionality if a0 becomes 2a0 then b0 will become 2b0 and c will be 2c0 . But i know i am wrong dont know the reason why . I am not understanding actually .


Doubt 3 : if " b and c are inversely proportional " so i assumed one eg the original value of b is 4 and c is 2 then b *c=8 so if we double the value of b to 2b =8 and new value of c will become half of the previous value 1/2*c i.e 1 so as to maintain constant ratio 8 between them . This is inversely proportional.(product of 2 variables equals to constant)

i know that x*y =k def of inverse but still not getting the right explanation why divide one and multiply other .

I am thinking this way if my first choice of value of x is 4 and y is 3 so in the first stage i know that 4*3=12 by defination of inverse . So i have to maintain this k (12) throughout my other values of x and y also .So if i try to change my x's value i.e 4 to 8 by multiplying 2 then i know that uptil now x has become 8 so the form is
8 * y=12 so the new value of y has to be 3/2 to maintain constant k so that previous value of y (3) has to be divided by 1/2 . thats why in inverse proportional we multiply and divide two values by the same factor. I hope my intution is correct?

(MAIN)DOUBT 4: i dont understand how this whole thing "a/(b×c)= Constant" is working out? What is the meaning of this whole thing. I understand 2 variable proportionality but more than 3 cannot understand.

in the case of direct proportionality a/b=constant i have made a table and it satisfies the ratio
a | b
2 |4 constant ratio of 2
4 |8
8|16

but when it is more than 2 variables in the form of a/(b*c) = constant how the manupulation of values of variables takes place in compared to 2 variable ?

i did it in my way in the above in the doubt1 section but it is not working
i think this is the main key line "we keep values other than the two under considerations constant" that i am not understanding


THIS IS NO DOUBT:

Another thing i discovered in direct proportion if all the values of a/b gives a constant ratio=2


a | b
4| 2 a0 b0 ---->original value to start with
8 | 4 2a0=a1 2b0=b1
56 |28 7a1=a2 7b1=b2

i discover that the form is coming like this a0/b0= 2a0(i.e a1)/2b0(i.e b1)= 7a1(i.e a2)/7b1(i.e b2)= k(2) we can shorten it like this
a0/b0= 2a0/2b0= 14a0/14b0 =k

at first when i read the article i thought that every term would be like : a0/b0= 2a0(i.e a1)/2b0(i.e b1) =2a1(i.e a2)/2b1(i.e b2)=k
or

a0/b0= 3a0(i.e a1)/3b0(i.e b1)= 3a1(i.e a2)/3b1(i.e b2) =k

PLEASE READ---> that means at first i thought if i begin the original two values multiplying by a factor "x" then i also need to multiply the next 2 values i.e a1 and b1 to get a2 and b2 (from a1*x ->a2 & b1*x->b2) by "x" only but that is not the case as i have shown in this example


a0/b0= 2a0(i.e a1)/2b0(i.e b1)= 7a1(i.e a2)/7b1(i.e b2)


carefully see --->first two values are multiplied by 2 (a0*2->a1& b0*2->b1) then the next two values a1 and b1 are multiplied by 7 which is a different factor from the previous 2
(a1*7->a2 & b1*7->b2)

it is just that i have to make sure for each transition from a0*x to a1 &b0*x to b1 or from a1*x1 to a2 & b1*x1 to b2 each pair must have its own common factor so that when every pair is divided into simplest terms every pair will give k (constant ratio)
 

Dr.Peterson

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Please stop reading poorly-written explanations. If you have trouble following it, then find a better explanation. Better still, just work through a textbook, which will introduce ideas in order so that you can be expected to understand the prerequisites.

Here is one site that presents this material as part of an orderly progression: https://www.purplemath.com/modules/variatn.htm
 

Saumyojit

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i have read the ratio proportion part in this site also . Please once see my doubt i have written taking a long time please sir i beg of u. U are my only hope . please see my doubt especially the doubt with a/(b*c) =k .
 

Saumyojit

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sir if u dont help i will break down surely i had so much hop while writing this that u will help
 

Saumyojit

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@Dr.Peterson please once see my doubt and help i beg of u . i am waiting sir
 

Saumyojit

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Please stop reading poorly-written explanations. If you have trouble following it, then find a better explanation. Better still, just work through a textbook, which will introduce ideas in order so that you can be expected to understand the prerequisites.

Here is one site that presents this material as part of an orderly progression: https://www.purplemath.com/modules/variatn.htm
i dont know why it feels this is poorly written i have understand the direct proportional part but i am having problemn in the next part where a/b*c has been written . yes i have seen ur refernce sir . joint variation but that has not helped me in my doubts .
 

Dr.Peterson

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Primarily, I don't want to answer you point by point because you wrote too much at once. It exhausts me just to consider trying to answer you, particularly with all the implied subscripts.

But the page you quote simply is not right; it does not clearly say what needs to be said. In particular, I read just as far as what you quoted,

a/(b×c)= Constant (Since a is proportional to both b and c so all three variables change “together”​

and it's wrong. All three do not change together; b and c change inversely to one another, so that a and b might increase while c decreases. That is probably what you are objecting to. (What they mean is probably right, and may be explained better later; but they didn't say that here.)

So, again, if you don't understand it, it may not be your fault. They may be saying something unclearly or even incorrectly, and you are under no obligation to figure out what they mean! Move on to an explanation that you do understand.

I say the same thing to students about their teachers: Sometimes a teacher just doesn't communicate well to a particular student; their styles, or language, or whatever don't mesh. So you find another. (Sometimes that first teacher is me, other times I'm the second, and often I'm the tutor they end up with.)

At the very least, don't ask us what they meant; ask about your own understanding of the topic. I have no patience to untangle what someone else says.
 

Saumyojit

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I am sorry if i disturbed u but i generated a lot of doubts reading the article thats why i took my time to write this post. and i tried to understand it by myself but failed tbh . U are a great teacher and in explaining things to me in the most easiest way that why i respect u a lot sir . I have no tutors . first i try to understand myself then when i cannot i completly break down and stress too much . thats why i love this forum as it is helping me a lot in maths truly . and please support me in my journey sir or i will helpless in the future . I truly respect u sir from the bottom of my heart . U have helped me a lot .

"don't ask us what they meant; ask about your own understanding of the topic" sir i know what is direct proportional is but i was not understanding few things like u mentioned " a/(b×c)= Constant (Since a is proportional to both b and c so all three variables change “together”

this term i could not understand what it meant actually i also dont know whther they are wrong or not because when i am reading something new i have to completely beleive it at first time BUT

I discovered that when a and b are directly proprotionately changing then the other value c has to be constant. so i got to know by what they are trying to mean by saying "we keep values other than the two under considerations constant "
 

Saumyojit

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From the link that u referred this eg :
If y varies jointly as x and z, and y = 5 when x = 3 and z = 4, then find y when x = 2 and z = 3.
y is coming 5/2


that means all the three variables are changing at the same time but in no proprotion to their old values
 

Dr.Peterson

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From the link that u referred this eg :
If y varies jointly as x and z, and y = 5 when x = 3 and z = 4, then find y when x = 2 and z = 3.
y is coming 5/2


that means all the three variables are changing at the same time but in no proprotion to their old values
How about if I try to explain it myself, rather than trying to explain someone else.

A key thing to keep in mind is that a statement like this tells how ONE variable, y, varies WHEN other variables, x and z, vary. It is not helpful to say that all three variables are changing in some way (though they are, of course, all changing). This is one of the errors in the page you cite, which make it worthless to try to analyze what it says. Although it is true in his example that a/(b×c)= Constant, the previous statement, a= (Constant) ×b×c, far better expresses the relationship, with a alone on the left. The idea is that if b increases while c is fixed, then a increases proportionally to b; and if c increases while b is fixed, then a increases proportionally to c. If b and c both increase, so will a; but if b increases while c decreases, the result depends on how each one changes.

In the example you quoted here, both x and z have decreased; one way to think about it (if you didn't have the formula) would be to imagine that one changes at a time. Since y is proportional to x, when x changes from 3 to 2 (multiplying by 2/3), y is multiplied by 2/3, resulting in 5*2/3 = 10/3. Then when z changes from 4 to 3 (multiplying by 3/4), y is multiplied by 3/4, resulting in 10/3 * 3/4 = 5/2.

Observe that we multiplied y by the ratio in which x changed, and by the ratio in which z changed. So we multiplied those two ratios. Yes, y did change in proportion to each of them!

But that's the slow way.

Using the method taught on Purplemath, "y varies jointly as x and z" means y = kxz; since 5 = k(3)(4), k = 5/12; so when x=2 and z=3, we have y = (5/12)(2)(3) = 5/2. Same result. Here we've combined the two changes into one formula. The result is the same as if one changed at a time.
 

Saumyojit

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@Dr.Peterson "but if b increases while c decreases " this has two cases
CASE 1: either b and c inversly proprotional keeping a constant suppose b=4 and c=18 and a =2 then in the next transition if i increase the value of b by 3 i.e b1=3*4=12 , so c will decrease by 1/3 c1 will be =18/3=6 but value of a will be 2

CASE 2: suppose b=3 c=4 a=5 then my k is 5/12 now if i increase b to b1= 4 and decrease c to c1= 2 to find the value of a1 we can deduce it from the formula so a1 will become 10/3

So yes the result depends on how each one changes.
 

Saumyojit

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In the example you quoted here, both x and z have decreased; one way to think about it (if you didn't have the formula) would be to imagine that one changes at a time. Since y is proportional to x, when x changes from 3 to 2 (multiplying by 2/3), y is multiplied by 2/3, resulting in 5*2/3 = 10/3. Then when z changes from 4 to 3 (multiplying by 3/4), y is multiplied by 3/4, resulting in 10/3 * 3/4 = 5/2.

Observe that we multiplied y by the ratio in which x changed, and by the ratio in which z changed. So we multiplied those two ratios. Yes, y did change in proportion to each of them!
That means when i know y is proportional to x first i change y to y1 in the same way x has been changed from the original value ,then at that moment i know z also has been changed so i know that y is also proportional to z so i change y1 another time according to as z initial value has been changed .

What i am not understanding is that x and z changes one time to x1 and z1 and if i think y also changes to y1 one time (formula way) but when i see ur above step it feels like y is changing two times ( one varying to x and another time varying to z).It is contradicting my intution of whether y changes one time or two time.
But this is the trick to ur method actually the variable(x or y or z ... in this case y) that will vary if i consider the non formula way the variable must change in proportion to both the other two variables so in inside that variable is changing two times but when the final value of that variable (y1 in this case) is found out it seems like it has gone through change one time only.

Am i able to interpret ur explanation correclty?
 

jonah2.0

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Beer soaked ramblings follow.
That means when i know y is proportional to x first i change y to y1 in the same way x has been changed from the original value ,then at that moment i know z also has been changed so i know that y is also proportional to z so i change y1 another time according to as z initial value has been changed .

What i am not understanding is that x and z changes one time to x1 and z1 and if i think y also changes to y1 one time (formula way) but when i see ur above step it feels like y is changing two times ( one varying to x and another time varying to z).It is contradicting my intution of whether y changes one time or two time.
But this is the trick to ur method actually the variable(x or y or z ... in this case y) that will vary if i consider the non formula way the variable must change in proportion to both the other two variables so in inside that variable is changing two times but when the final value of that variable (y1 in this case) is found out it seems like it has gone through change one time only.

Am i able to interpret ur explanation correclty?
Man that was exhausting and dizzying!
Do us all a favor and just post a screenshot of wherever you found your dilemma.
 

Saumyojit

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Beer soaked ramblings follow.

Man that was exhausting and dizzying!
Do us all a favor and just post a screenshot of wherever you found your dilemma.
u did not understand this then. This is no dilemma from the article that i had given .... i am confused in a particular area where Dr peterson have shown the non formula way .
Please read once again u will understand
what i am trying to say . Yeah i know its big but i need to layout everything that is going in my mind.
 

Dr.Peterson

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That means when i know y is proportional to x first i change y to y1 in the same way x has been changed from the original value ,then at that moment i know z also has been changed so i know that y is also proportional to z so i change y1 another time according to as z initial value has been changed .

What i am not understanding is that x and z changes one time to x1 and z1 and if i think y also changes to y1 one time (formula way) but when i see ur above step it feels like y is changing two times ( one varying to x and another time varying to z).It is contradicting my intution of whether y changes one time or two time.
But this is the trick to ur method actually the variable(x or y or z ... in this case y) that will vary if i consider the non formula way the variable must change in proportion to both the other two variables so in inside that variable is changing two times but when the final value of that variable (y1 in this case) is found out it seems like it has gone through change one time only.

Am i able to interpret ur explanation correclty?
I think so.

It just works out that you get the same result whether they change at the same time or sequentially (in either order). That is not true of everything (for example, it is not true of 3D motions), but is true of proportionality. Knowing that, you can now think of them changing at the same time: in my example, x is multiplied by 2/3 while z is multiplied by 3/4, so y is multiplied by 2/3 * 3/4 = 1/2.
 

Saumyojit

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I think so.

It just works out that you get the same result whether they change at the same time or sequentially (in either order). That is not true of everything (for example, it is not true of 3D motions), but is true of proportionality. Knowing that, you can now think of them changing at the same time: in my example, x is multiplied by 2/3 while z is multiplied by 3/4, so y is multiplied by 2/3 * 3/4 = 1/2.
in the non formula way it is like y has to obey and vary proportionately both according to the other variables before reaching its final value in each transition .
can u check if my explantion to myself is correct in doubt 3 section ....
 

Dr.Peterson

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in the non formula way it is like y has to obey and vary proportionately both according to the other variables before reaching its final value in each transition .
can u check if my explantion to myself is correct in doubt 3 section ....
Doubt 3 : if " b and c are inversely proportional " so i assumed one eg the original value of b is 4 and c is 2 then b *c=8 so if we double the value of b to 2b =8 and new value of c will become half of the previous value 1/2*c i.e 1 so as to maintain constant ratio 8 between them . This is inversely proportional.(product of 2 variables equals to constant)

i know that x*y =k def of inverse but still not getting the right explanation why divide one and multiply other .

I am thinking this way if my first choice of value of x is 4 and y is 3 so in the first stage i know that 4*3=12 by defination of inverse . So i have to maintain this k (12) throughout my other values of x and y also .So if i try to change my x's value i.e 4 to 8 by multiplying 2 then i know that uptil now x has become 8 so the form is 8 * y=12 so the new value of y has to be 3/2 to maintain constant k so that previous value of y (3) has to be divided by 1/2 . thats why in inverse proportional we multiply and divide two values by the same factor. I hope my intuition is correct?
If xy = k, then y = k/x. That's the easiest way to see it. Alternatively, in terms of xy = k, if we multiply one number by n and the other by 1/n (that is, divide by n), it should be clear that (nx)(y/n) = xy(n/n) = xy = k, so the relationship is still true. Doubling one number and halving the other results in the same product.

Your writing is hard to follow, but I suppose you are not wrong.
 

Saumyojit

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If xy = k, then y = k/x. That's the easiest way to see it.
if original value of x is 4 then y is 2 then 4*2=8 so i have update my x to 8(i.e x1) by doubling now i know to maintain ratio of 8 i need to divide by y by 2 so that what was i saying ...ok
 
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