Luke had ³₄ as many beads as Vaidev. Vaidev had ⁴₇ as many beads as Fiona. After Luke and Fiona gave 14 beads to Vaidev, Vaidev had thrice as many beads as Luke. Fiona then had the same number of beads as Vaidev.

1. What was the ratio of the number of beads Fiona had to the number of beads Luke had in the beginning? Answer in simplest form.
2. How many beads did Vaidev have in the end?

Luke	Vaidev	Fiona
3	4
	4	7
3	4	7

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

Ratio of the number of beads that Fiona had to the number of beads that Luke had in the beginning = 7 : 3

	Luke	Vaidev	Fiona	Total
Before	3x1 = 3 u	4x1 = 4 u	7x1 = 7 u	14x1 = 14 u
Change	- 14	+ 28	- 14
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 Vaidev received from Luke and Fiona
= 2 x 14
= 28

Number of beads that Vaidev received from Luke and Fiona
= 2 x 1 u
= 2 u

2 u = 14

1 u = 14 ÷ 2 = 7

Number of beads that Ashley had in the end
= 6 u
= 6 x 7
= 42