FDP (ED5) CH6 P99 Q9 - What is the ratio of x:y:z?

[jasonfko]

Hi,

I was looking at the answer key for this question and wasn't able to figure how it moved from one step to the other:

x + y = 2z
x = 2z - y
x/y = (2z-y) / y
x/y = 2z/y - 1 [not sure how it got to this step]

Can someone explain it to me? Thanks.

[RonPurewal]

x + y = 2z

... subtract y from both sides ...

x = 2z - y

... divide both sides by y ...

x/y = (2z-y) / y

... divide out the right-hand fraction ...

x/y = 2z/y - 1

the last step is how fractions can be broken up when there's more than one term added/subtracted in the numerator.
you may be familiar with this rule in other settings; for instance, you may already realize that, say, (15x + 18y)/3 is the same as 5x + 6y.
the reason why that works is that it's 15x/3 + 18y/3 -- i.e., you divide each term by the 3 that's on the bottom.
so, here, that's 2z/y - y/y. because y/y is just 1, you get what you get up there.

if you don't like that, you can always rewrite (2z - y)/y as the product of (2z - y) and 1/y, and then distribute, if you are ok with the whole distributing thing.
that's much more annoying than just learning to divide out these expressions, though.