If x is directly proportional to y, y is directly proportional to z and z is directly proportional to x, find the product of three variations constants.
Answers
Answer:
By definition of proportionality,
If x is directly proportional to y,
x α y
∴x=Ky .......Equation 1
where K is constant of proprtionality
Now,
Given that y=2,x=4
putting the above values in Equation 1 we get,
4=K×2
∴K=2
Answer:
If x is directly proportional to y and also inversely proportional to z, can I write x is directly proportional to y/z? How?
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What does this statement mean, exactly?
x is directly proportional to y .
It means that when all other terms which affect x are kept constant, x changes at a rate which is a constant multiple of the rate at which y changes.
You may have been taught to directly write
x∝y⟹x=ky
when you are told that x is directly proportional to y . But look at the statement with this new knowledge, which I shall repeat. When all other terms which affect x (in this case, z ) are kept constant, x changes at a rate which is constant multiple of the rate at which y changes.
Obviously, the constant multiple must be dependent on those other terms which affect x (i.e. the constant multiple depends on z ). From the question, it is clear that z is one of those other terms. Hence, the constant of proportionality k must somehow depend on z . You must write it like this.
x∝y⟹x=f(z)×y
Notice that I have simply replaced the constant of proportionality by a function of z .
The next piece of information tells us that
x is inversely proportional to z .
Again, this actually means that when all other terms which affect x (in this case, y ) are kept constant, x changes at a rate which is constant multiple of the rate at which 1z changes.
x=f(z)×y
This time, y is a constant. And x is inversely related to z . Therefore, f must be some kind of a reciprocal function!
f(z)=p×1z
Here, p is some constant. (The ultimate constant of this proportionality.)
⟹x=p×1z×y
⟹x=p×yz
⟹x∝yz
Hope this makes it clear.