Dana and Zara had a total of 154 buttons. Zara gave ¹₃ of her buttons to Dana. In return, Dana gave ¹₈ of the total number of buttons that she had to Zara. In the end, each girl had the same number of buttons. How many buttons did Dana have at first?

	Dana	Zara	Total
Before 1	?	3 u	154
Change 1	+ 1 u	- 1 u
After 1	88	2 u	154
Before 2	8 p	2 u	154
Change 2	- 1 p	+ 1 p
After 2	7 p (77)	77	154

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

Number of buttons that Zara and Dana each had in the end is the same.

Number of buttons that Dana had in the end
= 154 ÷ 2
= 77

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

7 p = 77

1 p = 77 ÷ 7 = 11

Number of buttons that Dana had after receiving some buttons from Zara
= 8 p
= 8 x 11
= 88

Number of buttons that Zara had after giving to Dana
= 154 - 88
= 66

2 u = 66

1 u = 66 ÷ 2 = 33

Number of buttons that Zara had at first
= 3 u
= 3 x 33
= 99

Number of buttons that Dana had at first
= 154 - 99
= 55

Answer(s): 55