Grace had 50% more buttons than Avery. Hazel had 20% fewer buttons than Grace. Grace and Avery gave Hazel some buttons in the ratio of 4 : 5. As a result, Hazel had 25% more buttons than before. Given that Hazel had 160 more buttons than Avery in the end, how many buttons did Grace give to Hazel?

Grace	Avery	Hazel
3x5	2x5
5x3		4x3
15	10	12

100% + 50% = 150%
150% = ¹⁵⁰₁₀₀ = ³₂

Grace : Avery = 3 : 2

100% - 20% = 80%
80% = ⁸⁰₁₀₀ = ⁴₅

Grace : Hazel = 5 : 4

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

	Grace	Avery	Hazel
Before	15x3 =  45 u	10x3 = 30 u	12x3 = 36 u
Change			+ 3x3 = + 9 u
Comparing the changes in the number of buttons	- 4x1 = - 4 u	- 5x1 = - 5 u	+ 9x1 = + 9 u
After	41 u	25 u	15x3 = 45 u

25% x 12 
= ²⁵₁₀₀ x 12 
= 3 

The total number of buttons that Grace and Avery gave to Hazel is repeated.  Make the total number of buttons that Grace and Avery gave to Hazel the same.  LCM of 9 and 3 is 9.

Number of buttons that Hazel had more than Avery 
= 45 u - 25 u
= 20 u

20 u = 160

1 u = 160 ÷ 20 = 8

Number of buttons that Grace gave to Hazel
= 4 u
= 4 x 8
= 32

Answer(s): 32