Isaac had 40% more buttons than Avery. Lucas had 20% fewer buttons than Isaac. Isaac and Avery gave Lucas some buttons in the ratio of 3 : 4. As a result, Lucas had 50% more buttons than before. Given that Avery had 100 less buttons than Lucas in the end, how many buttons did Avery give to Lucas?

Isaac	Avery	Lucas
7x5	5x5
5x7		4x7
35	25	28

100% + 40% = 140%
140% = ¹⁴⁰₁₀₀ = ⁷₅

Isaac : Avery = 7 : 5

100% - 20% = 80%
80% = ⁸⁰₁₀₀ = ⁴₅

Isaac : Lucas = 5 : 4

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

	Isaac	Avery	Lucas
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 Isaac and Avery gave to Lucas is repeated.  Make the total number of buttons that Isaac and Avery gave to Lucas the same.  LCM of 7 and 14 is 14.

Number of buttons that Avery had less than Lucas 
= 42 u - 17 u
= 25 u

25 u = 100

1 u = 100 ÷ 25 = 4

Number of buttons that Avery gave to Lucas
= 8 u
= 8 x 4
= 32

Answer(s): 32