Ahmad had ¹₂ as many beads as Archie. Archie had ²₃ as many beads as Betty. After Ahmad and Betty gave 12 beads to Archie, Archie had four times as many beads as Ahmad. Betty then had the same number of beads as Archie.

1. What was the ratio of the number of beads Betty had to the number of beads Ahmad had in the beginning? Answer in simplest form.
2. How many beads did Archie have in the end?

Ahmad	Archie	Betty
1	2
	2	3
1	2	3

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

Ratio of the number of beads that Betty had to the number of beads that Ahmad had in the beginning = 3 : 1

	Ahmad	Archie	Betty	Total
Before	1x3 = 3 u	2x3 = 6 u	3x3 = 9 u	6x3 = 18 u
Change	- 12	+ 24	- 12
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 Archie received from Ahmad and Betty
= 2 x 12
= 24

Number of beads that Archie received from Ahmad and Betty
= 2 x 1 u
= 2 u

2 u = 12

1 u = 12 ÷ 2 = 6

Number of beads that Ashley had in the end
= 8 u
= 8 x 6
= 48