Rebecca had 50% more beads than Claire. Christopher had 20% fewer beads than Rebecca. Rebecca and Claire gave Christopher some beads in the ratio of 3 : 5. As a result, Christopher had 50% more beads than before. Given that Christopher had 564 more beads than Claire in the end, how many beads did Rebecca give to Christopher?

Rebecca	Claire	Christopher
3x5	2x5
5x3		4x3
15	10	12

100% + 50% = 150%
150% = ¹⁵⁰₁₀₀ = ³₂

Rebecca : Claire = 3 : 2

100% - 20% = 80%
80% = ⁸⁰₁₀₀ = ⁴₅

Rebecca : Christopher = 5 : 4

The number of beads that Rebecca had at first is repeated. Make the number of beads that Rebecca had at first the same.  LCM of 3 and 5 is 15.

	Rebecca	Claire	Christopher
Before	15x4 =  60 u	10x4 = 40 u	12x4 = 48 u
Change			+ 6x4 = + 24 u
Comparing the changes in the number of beads	- 3x3 = - 9 u	- 5x3 = - 15 u	+ 8x3 = + 24 u
After	51 u	25 u	18x4 = 72 u

50% x 12 
= ⁵⁰₁₀₀ x 12 
= 6 

The total number of beads that Rebecca and Claire gave to Christopher is repeated.  Make the total number of beads that Rebecca and Claire gave to Christopher the same.  LCM of 8 and 6 is 24.

Number of beads that Christopher had more than Claire 
= 72 u - 25 u
= 47 u

47 u = 564

1 u = 564 ÷ 47 = 12

Number of beads that Rebecca gave to Christopher
= 9 u
= 9 x 12
= 108

Answer(s): 108