Gwen and Vanessa had a total of 48 cards. Vanessa gave ¹₅ of her cards to Gwen. In return, Gwen gave ¹₄ of the total number of cards that she had to Vanessa. In the end, each girl had the same number of cards. How many cards did Gwen have at first?

	Gwen	Vanessa	Total
Before 1	?	5 u	48
Change 1	+ 1 u	- 1 u
After 1	32	4 u	48
Before 2	4 p	4 u	48
Change 2	- 1 p	+ 1 p
After 2	3 p (24)	24	48

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

Number of cards that Vanessa and Gwen each had in the end is the same.

Number of cards that Gwen had in the end
= 48 ÷ 2
= 24

Number of cards that Gwen had in the end = 3 p

3 p = 24

1 p = 24 ÷ 3 = 8

Number of cards that Gwen had after receiving some cards from Vanessa
= 4 p
= 4 x 8
= 32

Number of cards that Vanessa had after giving to Gwen
= 48 - 32
= 16

4 u = 16

1 u = 16 ÷ 4 = 4

Number of cards that Vanessa had at first
= 5 u
= 5 x 4
= 20

Number of cards that Gwen had at first
= 48 - 20
= 28

Answer(s): 28