Benjamin had 50% more coins than Claire. Emma had 20% fewer coins than Benjamin. Benjamin and Claire gave Emma some coins in the ratio of 4 : 5. As a result, Emma had 50% more coins than before. Given that Claire had 408 less coins than Emma in the end, how many coins did Claire give to Emma?

Benjamin	Claire	Emma
3x5	2x5
5x3		4x3
15	10	12

100% + 50% = 150%
150% = ¹⁵⁰₁₀₀ = ³₂

Benjamin : Claire = 3 : 2

100% - 20% = 80%
80% = ⁸⁰₁₀₀ = ⁴₅

Benjamin : Emma = 5 : 4

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

	Benjamin	Claire	Emma
Before	15x3 =  45 u	10x3 = 30 u	12x3 = 36 u
Change			+ 6x3 = + 18 u
Comparing the changes in the number of coins	- 4x2 = - 8 u	- 5x2 = - 10 u	+ 9x2 = + 18 u
After	37 u	20 u	18x3 = 54 u

50% x 12 
= ⁵⁰₁₀₀ x 12 
= 6 

The total number of coins that Benjamin and Claire gave to Emma is repeated.  Make the total number of coins that Benjamin and Claire gave to Emma the same.  LCM of 9 and 6 is 18.

Number of coins that Claire had less than Emma 
= 54 u - 20 u
= 34 u

34 u = 408

1 u = 408 ÷ 34 = 12

Number of coins that Claire gave to Emma
= 10 u
= 10 x 12
= 120

Answer(s): 120