Luke had 50% more coins than Mason. Henry had 20% fewer coins than Luke. Luke and Mason gave Henry some coins in the ratio of 4 : 5. As a result, Henry had 50% more coins than before. Given that Henry had 204 more coins than Mason in the end, how many coins did Luke give to Henry?

Luke	Mason	Henry
3x5	2x5
5x3		4x3
15	10	12

100% + 50% = 150%
150% = ¹⁵⁰₁₀₀ = ³₂

Luke : Mason = 3 : 2

100% - 20% = 80%
80% = ⁸⁰₁₀₀ = ⁴₅

Luke : Henry = 5 : 4

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

	Luke	Mason	Henry
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 Luke and Mason gave to Henry is repeated.  Make the total number of coins that Luke and Mason gave to Henry the same.  LCM of 9 and 6 is 18.

Number of coins that Henry had more than Mason 
= 54 u - 20 u
= 34 u

34 u = 204

1 u = 204 ÷ 34 = 6

Number of coins that Luke gave to Henry
= 8 u
= 8 x 6
= 48

Answer(s): 48