**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)

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)