Dana and Penelope had a total of 84 cards. Penelope gave ¹₄ of her cards to Dana. In return, Dana gave ¹₈ of the total number of cards that she had to Penelope. In the end, each girl had the same number of cards. How many cards did Dana have at first?

	Dana	Penelope	Total
Before 1	?	4 u	84
Change 1	+ 1 u	- 1 u
After 1	48	3 u	84
Before 2	8 p	3 u	84
Change 2	- 1 p	+ 1 p
After 2	7 p (42)	42	84

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

Number of cards that Penelope and Dana each had in the end is the same.

Number of cards that Dana had in the end
= 84 ÷ 2
= 42

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

7 p = 42

1 p = 42 ÷ 7 = 6

Number of cards that Dana had after receiving some cards from Penelope
= 8 p
= 8 x 6
= 48

Number of cards that Penelope had after giving to Dana
= 84 - 48
= 36

3 u = 36

1 u = 36 ÷ 3 = 12

Number of cards that Penelope had at first
= 4 u
= 4 x 12
= 48

Number of cards that Dana had at first
= 84 - 48
= 36

Answer(s): 36