Zane had ³₄ as many beads as Julian. Julian had ⁴₇ as many beads as Diana. After Zane and Diana gave 28 beads to Julian, Julian had thrice as many beads as Zane. Diana then had the same number of beads as Julian.

1. What was the ratio of the number of beads Diana had to the number of beads Zane had in the beginning? Answer in simplest form.
2. How many beads did Julian have in the end?

Zane	Julian	Diana
3	4
	4	7
3	4	7

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

Ratio of the number of beads that Diana had to the number of beads that Zane had in the beginning = 7 : 3

	Zane	Julian	Diana	Total
Before	3x1 = 3 u	4x1 = 4 u	7x1 = 7 u	14x1 = 14 u
Change	- 28	+ 56	- 28
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 Julian received from Zane and Diana
= 2 x 28
= 56

Number of beads that Julian received from Zane and Diana
= 2 x 1 u
= 2 u

2 u = 28

1 u = 28 ÷ 2 = 14

Number of beads that Ashley had in the end
= 6 u
= 6 x 14
= 84