Cindy and Xuan had a total of 60 buttons. Xuan gave ¹₃ of her buttons to Cindy. In return, Cindy gave ¹₆ 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 Cindy have at first?

	Cindy	Xuan	Total
Before 1	?	3 u	60
Change 1	+ 1 u	- 1 u
After 1	36	2 u	60
Before 2	6 p	2 u	60
Change 2	- 1 p	+ 1 p
After 2	5 p (30)	30	60

Since Xuan gave some buttons to Cindy and Cindy 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 Cindy each had in the end is the same.

Number of buttons that Cindy had in the end
= 60 ÷ 2
= 30

Number of buttons that Cindy had in the end = 5 p

5 p = 30

1 p = 30 ÷ 5 = 6

Number of buttons that Cindy had after receiving some buttons from Xuan
= 6 p
= 6 x 6
= 36

Number of buttons that Xuan had after giving to Cindy
= 60 - 36
= 24

2 u = 24

1 u = 24 ÷ 2 = 12

Number of buttons that Xuan had at first
= 3 u
= 3 x 12
= 36

Number of buttons that Cindy had at first
= 60 - 36
= 24

Answer(s): 24