Peter had ³₄ as many beads as Dylan. Dylan had ⁴₇ as many beads as Natalie. After Peter and Natalie gave 22 beads to Dylan, Dylan had thrice as many beads as Peter. Natalie then had the same number of beads as Dylan.

1. What was the ratio of the number of beads Natalie had to the number of beads Peter had in the beginning? Answer in simplest form.
2. How many beads did Dylan have in the end?

Peter	Dylan	Natalie
3	4
	4	7
3	4	7

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

Ratio of the number of beads that Natalie had to the number of beads that Peter had in the beginning = 7 : 3

	Peter	Dylan	Natalie	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 Dylan received from Peter and Natalie
= 2 x 22
= 44

Number of beads that Dylan received from Peter and Natalie
= 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