Xuan and Opal had a total of 28 buttons. Opal gave ¹₃ of her buttons to Xuan. In return, Xuan gave ¹₈ of the total number of buttons that she had to Opal. In the end, each girl had the same number of buttons. How many buttons did Xuan have at first?

	Xuan	Opal	Total
Before 1	?	3 u	28
Change 1	+ 1 u	- 1 u
After 1	16	2 u	28
Before 2	8 p	2 u	28
Change 2	- 1 p	+ 1 p
After 2	7 p (14)	14	28

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

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

Number of buttons that Xuan had in the end
= 28 ÷ 2
= 14

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

7 p = 14

1 p = 14 ÷ 7 = 2

Number of buttons that Xuan had after receiving some buttons from Opal
= 8 p
= 8 x 2
= 16

Number of buttons that Opal had after giving to Xuan
= 28 - 16
= 12

2 u = 12

1 u = 12 ÷ 2 = 6

Number of buttons that Opal had at first
= 3 u
= 3 x 6
= 18

Number of buttons that Xuan had at first
= 28 - 18
= 10

Answer(s): 10