Nora and Billy collect pencils and buttons. Billy has 45 less pencils than buttons. Billy has 2 times the number of pencils Nora has, and Nora has 20% more buttons than Billy. After Nora gives Billy all of her pencils and Billy gives Nora 60% of his buttons, Billy has 15 less buttons than pencils. Find the number of buttons Nora has at first.

Nora		Billy
Buttons	Pencils	Buttons	Pencils
2.4 u + 54	1 u	2 u + 45	2 u
+ 1.2 u + 27	- 1 u	- 1.2 u - 27	+ 1 u
3.6 u + 81	0	0.8 u + 18	3 u

100%+ 20% = 120% 

120% x 2 u
= ¹²⁰₁₀₀ x 2 u
= 2.4 u

120% x 45
= ¹²⁰₁₀₀ x 45
= 54 

60% x 2 u
= ⁶⁰₁₀₀ x 2 u
= 1.2 u  

60% x 45 
= ⁶⁰₁₀₀ x 45
= 27 

If another 15 buttons are given to Billy, he will have equal number of buttons and pencils in the end.
3 u = 0.8 u + 18 + 15
3 u - 0.8 u = 33
2.2 u = 33

1 u = 33 ÷ 2.2 = 15

Number of buttons that Nora has at first
= 2.4 u + 54
= 2.4 x 15 + 54
= 36 + 54
= 90
 
Answer(s): 90