Luke had 50% more marbles than Liam. Lily had 20% fewer marbles than Luke. Luke and Liam gave Lily some marbles in the ratio of 3 : 5. As a result, Lily had 50% more marbles than before. Given that Lily had 329 more marbles than Liam in the end, how many marbles did Luke give to Lily?

Luke	Liam	Lily
3x5	2x5
5x3		4x3
15	10	12

100% + 50% = 150%
150% = ¹⁵⁰₁₀₀ = ³₂

Luke : Liam = 3 : 2

100% - 20% = 80%
80% = ⁸⁰₁₀₀ = ⁴₅

Luke : Lily = 5 : 4

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

	Luke	Liam	Lily
Before	15x4 =  60 u	10x4 = 40 u	12x4 = 48 u
Change			+ 6x4 = + 24 u
Comparing the changes in the number of marbles	- 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 marbles that Luke and Liam gave to Lily is repeated.  Make the total number of marbles that Luke and Liam gave to Lily the same.  LCM of 8 and 6 is 24.

Number of marbles that Lily had more than Liam 
= 72 u - 25 u
= 47 u

47 u = 329

1 u = 329 ÷ 47 = 7

Number of marbles that Luke gave to Lily
= 9 u
= 9 x 7
= 63

Answer(s): 63