Bryan had ¹₂ as many cards as Howard. Howard had ²₃ as many cards as Diana. After Bryan and Diana gave 28 cards to Howard, Howard had four times as many cards as Bryan. Diana then had the same number of cards as Howard.

1. What was the ratio of the number of cards Bryan had to the number of cards Diana had in the beginning? Answer in simplest form.
2. How many cards did Howard have in the end?

Bryan	Howard	Diana
1	2
	2	3
1	2	3

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

Ratio of the number of cards that Bryan had to the number of cards that Diana had in the beginning = 1 : 3

	Bryan	Howard	Diana	Total
Before	1x3 = 3 u	2x3 = 6 u	3x3 = 9 u	6x3 = 18 u
Change	- 28	+ 56	- 28
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 Howard received from Bryan and Diana
= 2 x 28
= 56

Number of cards that Howard received from Bryan and Diana
= 2 x 1 u
= 2 u

2 u = 28

1 u = 28 ÷ 2 = 14

Number of cards that Ashley had in the end
= 8 u
= 8 x 14
= 112