Zoe and Dana had a total of 80 coins. Dana gave ¹₃ of her coins to Zoe. In return, Zoe gave ¹₅ of the total number of coins that she had to Dana. In the end, each girl had the same number of coins. How many coins did Zoe have at first?

	Zoe	Dana	Total
Before 1	?	3 u	80
Change 1	+ 1 u	- 1 u
After 1	50	2 u	80
Before 2	5 p	2 u	80
Change 2	- 1 p	+ 1 p
After 2	4 p (40)	40	80

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

Number of coins that Dana and Zoe each had in the end is the same.

Number of coins that Zoe had in the end
= 80 ÷ 2
= 40

Number of coins that Zoe had in the end = 4 p

4 p = 40

1 p = 40 ÷ 4 = 10

Number of coins that Zoe had after receiving some coins from Dana
= 5 p
= 5 x 10
= 50

Number of coins that Dana had after giving to Zoe
= 80 - 50
= 30

2 u = 30

1 u = 30 ÷ 2 = 15

Number of coins that Dana had at first
= 3 u
= 3 x 15
= 45

Number of coins that Zoe had at first
= 80 - 45
= 35

Answer(s): 35