Roshel and Xuan had a total of 168 buttons. Xuan gave ¹₃ of her buttons to Roshel. In return, Roshel gave ¹₈ of the total number of buttons that she had to Xuan. In the end, each girl had the same number of buttons. How many buttons did Roshel have at first?

	Roshel	Xuan	Total
Before 1	?	3 u	168
Change 1	+ 1 u	- 1 u
After 1	96	2 u	168
Before 2	8 p	2 u	168
Change 2	- 1 p	+ 1 p
After 2	7 p (84)	84	168

Since Xuan gave some buttons to Roshel and Roshel then gave some buttons to Xuan, it is an internal transfer of buttons between the two girls. So, the total number of buttons remains unchanged.

Number of buttons that Xuan and Roshel each had in the end is the same.

Number of buttons that Roshel had in the end
= 168 ÷ 2
= 84

Number of buttons that Roshel had in the end = 7 p

7 p = 84

1 p = 84 ÷ 7 = 12

Number of buttons that Roshel had after receiving some buttons from Xuan
= 8 p
= 8 x 12
= 96

Number of buttons that Xuan had after giving to Roshel
= 168 - 96
= 72

2 u = 72

1 u = 72 ÷ 2 = 36

Number of buttons that Xuan had at first
= 3 u
= 3 x 36
= 108

Number of buttons that Roshel had at first
= 168 - 108
= 60

Answer(s): 60