Aubrey had 50% more buttons than Aria. Hannah had 20% fewer buttons than Aubrey. Aubrey and Aria gave Hannah some buttons in the ratio of 1 : 3. As a result, Hannah had 25% more buttons than before. Given that Aria had 145 less buttons than Hannah in the end, how many buttons did Aria give to Hannah?

Aubrey	Aria	Hannah
3x5	2x5
5x3		4x3
15	10	12

100% + 50% = 150%
150% = ¹⁵⁰₁₀₀ = ³₂

Aubrey : Aria = 3 : 2

100% - 20% = 80%
80% = ⁸⁰₁₀₀ = ⁴₅

Aubrey : Hannah = 5 : 4

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

	Aubrey	Aria	Hannah
Before	15x4 =  60 u	10x4 = 40 u	12x4 = 48 u
Change			+ 3x4 = + 12 u
Comparing the changes in the number of buttons	- 1x3 = - 3 u	- 3x3 = - 9 u	+ 4x3 = + 12 u
After	57 u	31 u	15x4 = 60 u

25% x 12 
= ²⁵₁₀₀ x 12 
= 3 

The total number of buttons that Aubrey and Aria gave to Hannah is repeated.  Make the total number of buttons that Aubrey and Aria gave to Hannah the same.  LCM of 4 and 3 is 12.

Number of buttons that Aria had less than Hannah 
= 60 u - 31 u
= 29 u

29 u = 145

1 u = 145 ÷ 29 = 5

Number of buttons that Aria gave to Hannah
= 9 u
= 9 x 5
= 45

Answer(s): 45