Mark had ³₄ as many cards as Ivan. Ivan had ⁴₇ as many cards as Roshel. After Mark and Roshel gave 18 cards to Ivan, Ivan had thrice as many cards as Mark. Roshel then had the same number of cards as Ivan.

1. What was the ratio of the number of cards Mark had to the number of cards Roshel had in the beginning? Answer in simplest form.
2. How many cards did Ivan have in the end?

Mark	Ivan	Roshel
3	4
	4	7
3	4	7

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

Ratio of the number of cards that Mark had to the number of cards that Roshel had in the beginning = 3 : 7

	Mark	Ivan	Roshel	Total
Before	3x1 = 3 u	4x1 = 4 u	7x1 = 7 u	14x1 = 14 u
Change	- 18	+ 36	- 18
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 Ivan received from Mark and Roshel
= 2 x 18
= 36

Number of cards that Ivan received from Mark and Roshel
= 2 x 1 u
= 2 u

2 u = 18

1 u = 18 ÷ 2 = 9

Number of cards that Ashley had in the end
= 6 u
= 6 x 9
= 54