what is the difference between saying that one quantity is proportional to another and saying it is equal to a

Answer 1

I think you mean to ask what is the difference between the two following statements:

1) X is proportional to Y.

2) X equals Y.

If this is your question, then here is the answer:

When X is proportional to Y, that means that as X gets bigger, Y gets bigger. When X gets smaller, Y gets smaller. And, one can predict how much bigger or smaller Y will be.

You can state the difference mathematically in the following ways:

1) y = zx

2) y = x

The first statement shows a proportion: If you know x, you can determine y by multiplying x by certain number. This number (z) can be any nonzero number.

Hope this helps.

Answer 2

I think the intended question is
"what is the difference between saying that one quantity is proportional to another and saying it is equal to a constant times another?"
If that is the question, the difference is that "proportional" doesn't state what the proportionality factor is, and allows the possibility that the factor can change, such as due to variations in other conditions. In contrast, "constant" means the factor is an unchanging number.