Jean and Penelope had a total of 96 cards. Penelope gave ¹₃ of her cards to Jean. In return, Jean gave ¹₇ 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 Jean have at first?

	Jean	Penelope	Total
Before 1	?	3 u	96
Change 1	+ 1 u	- 1 u
After 1	56	2 u	96
Before 2	7 p	2 u	96
Change 2	- 1 p	+ 1 p
After 2	6 p (48)	48	96

Since Penelope gave some cards to Jean and Jean 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 Jean each had in the end is the same.

Number of cards that Jean had in the end
= 96 ÷ 2
= 48

Number of cards that Jean had in the end = 6 p

6 p = 48

1 p = 48 ÷ 6 = 8

Number of cards that Jean had after receiving some cards from Penelope
= 7 p
= 7 x 8
= 56

Number of cards that Penelope had after giving to Jean
= 96 - 56
= 40

2 u = 40

1 u = 40 ÷ 2 = 20

Number of cards that Penelope had at first
= 3 u
= 3 x 20
= 60

Number of cards that Jean had at first
= 96 - 60
= 36

Answer(s): 36