Barbara and Diana had a total of 140 cards. Diana gave ¹₃ of her cards to Barbara. In return, Barbara gave ¹₈ of the total number of cards that she had to Diana. In the end, each girl had the same number of cards. How many cards did Barbara have at first?

	Barbara	Diana	Total
Before 1	?	3 u	140
Change 1	+ 1 u	- 1 u
After 1	80	2 u	140
Before 2	8 p	2 u	140
Change 2	- 1 p	+ 1 p
After 2	7 p (70)	70	140

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

Number of cards that Diana and Barbara each had in the end is the same.

Number of cards that Barbara had in the end
= 140 ÷ 2
= 70

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

7 p = 70

1 p = 70 ÷ 7 = 10

Number of cards that Barbara had after receiving some cards from Diana
= 8 p
= 8 x 10
= 80

Number of cards that Diana had after giving to Barbara
= 140 - 80
= 60

2 u = 60

1 u = 60 ÷ 2 = 30

Number of cards that Diana had at first
= 3 u
= 3 x 30
= 90

Number of cards that Barbara had at first
= 140 - 90
= 50

Answer(s): 50