Ivan had ³₄ as many beads as Oliver. Oliver had ⁴₇ as many beads as Betty. After Ivan and Betty gave 28 beads to Oliver, Oliver had thrice as many beads as Ivan. Betty then had the same number of beads as Oliver.

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

Ivan	Oliver	Betty
3	4
	4	7
3	4	7

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

Ratio of the number of beads that Betty had to the number of beads that Ivan had in the beginning = 7 : 3

	Ivan	Oliver	Betty	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 Oliver received from Ivan and Betty
= 2 x 28
= 56

Number of beads that Oliver received from Ivan and Betty
= 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