Ben had ³₄ as many beads as Gabriel. Gabriel had ⁴₇ as many beads as Risa. After Ben and Risa gave 22 beads to Gabriel, Gabriel had thrice as many beads as Ben. Risa then had the same number of beads as Gabriel.

1. What was the ratio of the number of beads Risa had to the number of beads Ben had in the beginning? Answer in simplest form.
2. How many beads did Gabriel have in the end?

Ben	Gabriel	Risa
3	4
	4	7
3	4	7

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

Ratio of the number of beads that Risa had to the number of beads that Ben had in the beginning = 7 : 3

	Ben	Gabriel	Risa	Total
Before	3x1 = 3 u	4x1 = 4 u	7x1 = 7 u	14x1 = 14 u
Change	- 22	+ 44	- 22
After	1x2 = 2 u	3x2 = 6 u	3x2 = 6 u	7x2 = 14 u

The total number of beads at first and in the end remains unchanged. Make the total number of beads the same. LCM of 14 and 7 is 14.

Number of beads that Gabriel received from Ben and Risa
= 2 x 22
= 44

Number of beads that Gabriel received from Ben and Risa
= 2 x 1 u
= 2 u

2 u = 22

1 u = 22 ÷ 2 = 11

Number of beads that Ashley had in the end
= 6 u
= 6 x 11
= 66