Riordan had ¹₂ as many beads as Howard. Howard had ²₃ as many beads as Kimberly. After Riordan and Kimberly gave 14 beads to Howard, Howard had four times as many beads as Riordan. Kimberly then had the same number of beads as Howard.

1. What was the ratio of the number of beads Kimberly had to the number of beads Riordan had in the beginning? Answer in simplest form.
2. How many beads did Howard have in the end?

Riordan	Howard	Kimberly
1	2
	2	3
1	2	3

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

Ratio of the number of beads that Kimberly had to the number of beads that Riordan had in the beginning = 3 : 1

	Riordan	Howard	Kimberly	Total
Before	1x3 = 3 u	2x3 = 6 u	3x3 = 9 u	6x3 = 18 u
Change	- 14	+ 28	- 14
After	1x2 = 2 u	4x2 = 8 u	4x2 = 8 u	9x2 = 18 u

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

Number of beads that Howard received from Riordan and Kimberly
= 2 x 14
= 28

Number of beads that Howard received from Riordan and Kimberly
= 2 x 1 u
= 2 u

2 u = 14

1 u = 14 ÷ 2 = 7

Number of beads that Ashley had in the end
= 8 u
= 8 x 7
= 56