Reggie had ¹₂ as many cards as Ivan. Ivan had ²₃ as many cards as Olivia. After Reggie and Olivia gave 4 cards to Ivan, Ivan had four times as many cards as Reggie. Olivia then had the same number of cards as Ivan.

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

Reggie	Ivan	Olivia
1	2
	2	3
1	2	3

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

Ratio of the number of cards that Reggie had to the number of cards that Olivia had in the beginning = 1 : 3

	Reggie	Ivan	Olivia	Total
Before	1x3 = 3 u	2x3 = 6 u	3x3 = 9 u	6x3 = 18 u
Change	- 4	+ 8	- 4
After	1x2 = 2 u	4x2 = 8 u	4x2 = 8 u	9x2 = 18 u

The total number of cards at first and in the end remains unchanged. Make the total number of cards the same. LCM of 6 and 9 is 18.

Number of cards that Ivan received from Reggie and Olivia
= 2 x 4
= 8

Number of cards that Ivan received from Reggie and Olivia
= 2 x 1 u
= 2 u

2 u = 4

1 u = 4 ÷ 2 = 2

Number of cards that Ashley had in the end
= 8 u
= 8 x 2
= 16