Jason and Jian Ping each collect Vietnam War books. Initially, Jason had 6 times as many books as Jian Ping. After Jason gave Jian Ping some of his books for her birthday, the ratio of Jason's books to Jian Ping's was 9:5, and Jason had 12 more books than Jian Ping. How many Vietnam War books did Jian Ping have initially?

Let's use x for number of books Jason has and y for number of books Jian Ping has.

We know that Jason had 6 times as many books as Jian Ping. 
x=6y
We know that after Jason gave Jian Ping some of his books for her birthday, the ratio of Jason's books to Jian Ping's was 9:5. Let's use n for number of books that Jason gave to Jian Ping.
(x-n):(y+n)=9:5
We know that after Jason gave Jian Ping some of his books for her birthday, Jason had 12 more books than Jian Ping.
(x-n) - (y+n)=12

The equations we have are:
x=6y
(x-n):(y+n)=9:5
(x-n) - (y+n)=12
Let's multiply two outside numbers in second equation and put them on the left side and let's multiply two inside numbers and put them on the right side. Also, let's get rid of brackets in third equation.
x=6y
5*(x-n)=9*(y+n)
x-n - y-n=12
After some calculations we have:
x=6y
5x-5n=9y+9n
x-n - y-n=12
And
x=6y
5x-14n=9y
x-y-2n=12

We insert first equation into second and third one.
5*6y-14n=9y
6y-y-2n=12
We have
21y-14n=0 /:7
5y-2n=12
And
3y-2n=0
5y-2n=12 /*(-1)
And
3y-2n=0
-5y+2n=-12
Now we add both of these equations:
-2y=-12 /:(-2)
y=6
Now we can calculate the number of books that Jason had at the begining.
x=6y
x=6*6=36
Number of books that Jason gave to Jian Ping:
3*6-2n=0
18-2n=0
n=9

At the begining Jason had 36 books and Jian Ping had 6 books.