Sebastian had 50% more beads than Benjamin. Josiah had 20% fewer beads than Sebastian. Sebastian and Benjamin gave Josiah some beads in the ratio of 3 : 5. As a result, Josiah had 50% more beads than before. Given that Josiah had 423 more beads than Benjamin in the end, how many beads did Sebastian give to Josiah?

Sebastian	Benjamin	Josiah
3x5	2x5
5x3		4x3
15	10	12

100% + 50% = 150%
150% = ¹⁵⁰₁₀₀ = ³₂

Sebastian : Benjamin = 3 : 2

100% - 20% = 80%
80% = ⁸⁰₁₀₀ = ⁴₅

Sebastian : Josiah = 5 : 4

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

	Sebastian	Benjamin	Josiah
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 Sebastian and Benjamin gave to Josiah is repeated.  Make the total number of beads that Sebastian and Benjamin gave to Josiah the same.  LCM of 8 and 6 is 24.

Number of beads that Josiah had more than Benjamin 
= 72 u - 25 u
= 47 u

47 u = 423

1 u = 423 ÷ 47 = 9

Number of beads that Sebastian gave to Josiah
= 9 u
= 9 x 9
= 81

Answer(s): 81