A certain proportion

I need help explaining how to solve:

 12      36
---- = ----
3w 63

Could you solve this if instead of 3w it was x?

Well you can do just that:

12 36
---- = ----
x 63

Now cross multiply and go about it as usual. The answer is x = 21.

But, x is actually 3w. So 3w = 21. So w = 7.

This changing of variables is a very often used "trick" in mathematics.


Alternate solution.

Upon examining the proportion, we notice both the numbers in the first fraction are divisible by 3, and both numbers in the second fraction are divisible by 9.

12 36
---- = ----
3w 63

So, we can write it as follows and simplify both fractions (the right and left sides):

3 * 4 9 * 4
------ = -------
3w 9 * 7


4 4
--- = ---
w 7
You can't find a simpler proportion than that, obviously w must equal 7.
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