Henry had ³₄ as many cards as Jeremy. Jeremy had ⁴₇ as many cards as Gwen. After Henry and Gwen gave 6 cards to Jeremy, Jeremy had thrice as many cards as Henry. Gwen then had the same number of cards as Jeremy.

1. What was the ratio of the number of cards Henry had to the number of cards Gwen had in the beginning? Answer in simplest form.
2. How many cards did Jeremy have in the end?

Henry	Jeremy	Gwen
3	4
	4	7
3	4	7

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

Ratio of the number of cards that Henry had to the number of cards that Gwen had in the beginning = 3 : 7

	Henry	Jeremy	Gwen	Total
Before	3x1 = 3 u	4x1 = 4 u	7x1 = 7 u	14x1 = 14 u
Change	- 6	+ 12	- 6
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 Jeremy received from Henry and Gwen
= 2 x 6
= 12

Number of cards that Jeremy received from Henry and Gwen
= 2 x 1 u
= 2 u

2 u = 6

1 u = 6 ÷ 2 = 3

Number of cards that Ashley had in the end
= 6 u
= 6 x 3
= 18