Natalie and Gabriel collect buttons and cards. Gabriel has 65 more cards than buttons. Gabriel has 4 times the number of buttons Natalie has, and Natalie has 20% more cards than Gabriel. After Natalie gives Gabriel all of her buttons and Gabriel gives Natalie 60% of his cards, Gabriel has 42 more buttons than cards. Find the number of cards Natalie has at first.

Natalie		Gabriel
Cards	Buttons	Cards	Buttons
4.8 u + 78	1 u	4 u + 65	4 u
+ 2.4 u + 39	- 1 u	- 2.4 u - 39	+ 1 u
7.2 u + 117	0	1.6 u + 26	5 u

100% + 20% = 120% 

120% x 4 u
= ¹²⁰₁₀₀ x 4 u
= 4.8 u

120% x 65
= ¹²⁰₁₀₀ x 65
= 78 

60% x 4 u
= ⁶⁰₁₀₀ x 4 u
= 2.4 u  

60% x 65 
= ⁶⁰₁₀₀ x 65
= 39 

If another 42 cards are given to Gabriel, he will have equal number of cards and buttons in the end.
5 u = 1.6 u + 26 + 42
5 u - 1.6 u = 68
3.4 u = 68

1 u = 68 ÷ 3.4 = 20

Number of cards that Natalie has at first
= 4.8 u + 78
= 4.8 x 20 + 78
= 96 + 78
= 174
 
Answer(s): 174