Noah had 25% more coins than Ethan. Luke had 50% fewer coins than Noah. Noah and Ethan gave Luke some coins in the ratio of 1 : 3. As a result, Luke had 60% more coins than before. Given that Ethan had 108 less coins than Luke in the end, how many coins did Ethan give to Luke?

Noah	Ethan	Luke
5x2	4x2
2x5		1x5
10	8	5

100% + 25% = 125%
125% = ¹²⁵₁₀₀ = ⁵₄

Noah : Ethan = 5 : 4

100% - 50% = 50%
50% = ⁵⁰₁₀₀ = ¹₂

Noah : Luke = 2 : 1

The number of coins that Noah had at first is repeated. Make the number of coins that Noah had at first the same.  LCM of 5 and 2 is 10.

	Noah	Ethan	Luke
Before	10x4 =  40 u	8x4 = 32 u	5x4 = 20 u
Change			+ 3x4 = + 12 u
Comparing the changes in the number of coins	- 1x3 = - 3 u	- 3x3 = - 9 u	+ 4x3 = + 12 u
After	37 u	23 u	8x4 = 32 u

60% x 5 
= ⁶⁰₁₀₀ x 5 
= 3 

The total number of coins that Noah and Ethan gave to Luke is repeated.  Make the total number of coins that Noah and Ethan gave to Luke the same.  LCM of 4 and 3 is 12.

Number of coins that Ethan had less than Luke 
= 32 u - 23 u
= 9 u

9 u = 108

1 u = 108 ÷ 9 = 12

Number of coins that Ethan gave to Luke
= 9 u
= 9 x 12
= 108

Answer(s): 108