Cici and Amatina have a lot of stickers.?

Cici and Amatina have a lot of stickers. Cici had one-third as many as Amatina had, but then Amatina gave her six stickers, so now Cici has half as many as Amatina. How many stickers did each girl start with?

Ashley,

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Q: How many stickers did each girl start with?
A: Cici had 18, Amatina 54.

RATIONALE:

In order to translate the story into equations, we need to define the following variables:

Cs = Number of stickers Cici started with
Ce = Number of stickers Cici ended up with

As = Number of stickers Amatina started with
Ae = Number of stickers Amatina ended up with

Let's translate the whole story into equations:

At the start, Cici had one-third as many as Amatina had.
> Cs = As / 3

Then Amatina gave her six stickers.
> Ae = As - 6
> Ce = Cs + 6

So now Cici has half as many as Amatina.
> Ce = Ae / 2

So these are the 4 equations inferred:

Cs = As/3
Ce = Cs+6
Ae = As-6
Ce = Ae/2

Now we want to discover the number of stickers both had to start with. So the plan is to come up with 2 equations containing Cs and As (with the format Cs = xAs + y). We already have one (Cs = As/3), thus we need another one from the 3 other initial ones.

Ce = Cs+6
Ae = As-6
Ce = Ae/2

So let's start.

Given Ae = As - 6,
Ce = Ae/2
<=> Ce = (As-6)/2
<=> Ce = As/2 - 6/2
<=> Ce = As/2 - 3

Given Ce = Cs+6,
Ce = As/2 - 3
<=> Cs + 6 = As/2 - 3
<=> Cs = As/2 - 3 - 6
<=> Cs = As/2 - 9

So the 2nd equation is Cs = As/2 - 9

So we end up with the 2 following equations:

Cs = As/3
Cs = As/2 - 9

Let's resolve the first variable As.

Given Cs = As/3
Cs = As/2 - 9
<=> As/3 = As/2 - 9
<=> As/3 - As/2 = -9
<=> 2As/6 - 3As/6 = -9
<=> (2As - 3As)/6 = -9
<=> -As/6 = -9
<=> As/(-6) = -9
<=> As = (-9)(-6)
<=> As = 54

So now let's resolve the second variable Cs.

Given As = 54
Cs = As/3
<=> Cs = 54/3
<=> Cs = 18

SOLUTION:

Cs = 18
As = 54

Cici had 18 stickers to start with, Amatina 54.

VERIFICATION:

Let's double check, in case.

Cici had one-third as many as Amatina had.
> Right, because Cici has 18 (=54/3).

Amatina gave her six stickers
> so now Cici has 24 (=18+6) and Amatina 48 (=54-6)

So now Cici has half as many as Amatina.
> Right, because Cici has 24, which is hasl as much as Amatina's 48.

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