Lily had 40% more buttons than Ethan. Jackson had 20% fewer buttons than Lily. Lily and Ethan gave Jackson some buttons in the ratio of 3 : 4. As a result, Jackson had 50% more buttons than before. Given that Jackson had 225 more buttons than Ethan in the end, how many buttons did Lily give to Jackson?

Lily	Ethan	Jackson
7x5	5x5
5x7		4x7
35	25	28

100% + 40% = 140%
140% = ¹⁴⁰₁₀₀ = ⁷₅

Lily : Ethan = 7 : 5

100% - 20% = 80%
80% = ⁸⁰₁₀₀ = ⁴₅

Lily : Jackson = 5 : 4

The number of buttons that Lily had at first is repeated. Make the number of buttons that Lily had at first the same.  LCM of 7 and 5 is 35.

	Lily	Ethan	Jackson
Before	35x1 =  35 u	25x1 = 25 u	28x1 = 28 u
Change			+ 14x1 = + 14 u
Comparing the changes in the number of buttons	- 3x2 = - 6 u	- 4x2 = - 8 u	+ 7x2 = + 14 u
After	29 u	17 u	42x1 = 42 u

50% x 28 
= ⁵⁰₁₀₀ x 28 
= 14 

The total number of buttons that Lily and Ethan gave to Jackson is repeated.  Make the total number of buttons that Lily and Ethan gave to Jackson the same.  LCM of 7 and 14 is 14.

Number of buttons that Jackson had more than Ethan 
= 42 u - 17 u
= 25 u

25 u = 225

1 u = 225 ÷ 25 = 9

Number of buttons that Lily gave to Jackson
= 6 u
= 6 x 9
= 54

Answer(s): 54