John had ³₄ as many cards as Xavier. Xavier had ⁴₇ as many cards as Natalie. After John and Natalie gave 4 cards to Xavier, Xavier had thrice as many cards as John. Natalie then had the same number of cards as Xavier.

1. What was the ratio of the number of cards John had to the number of cards Natalie had in the beginning? Answer in simplest form.
2. How many cards did Xavier have in the end?

John	Xavier	Natalie
3	4
	4	7
3	4	7

(a)
The number of cards that Xavier is repeated.  

Ratio of the number of cards that John had to the number of cards that Natalie had in the beginning = 3 : 7

	John	Xavier	Natalie	Total
Before	3x1 = 3 u	4x1 = 4 u	7x1 = 7 u	14x1 = 14 u
Change	- 4	+ 8	- 4
After	1x2 = 2 u	3x2 = 6 u	3x2 = 6 u	7x2 = 14 u

The total number of cards at first and in the end remains unchanged. Make the total number of cards the same. LCM of 14 and 7 is 14.

Number of cards that Xavier received from John and Natalie
= 2 x 4
= 8

Number of cards that Xavier received from John and Natalie
= 2 x 1 u
= 2 u

2 u = 4

1 u = 4 ÷ 2 = 2

Number of cards that Ashley had in the end
= 6 u
= 6 x 2
= 12