Fanny and Barbara had a total of 80 buttons. Barbara gave ¹₆ of her buttons to Fanny. In return, Fanny gave ¹₅ of the total number of buttons that she had to Barbara. In the end, each girl had the same number of buttons. How many buttons did Fanny have at first?

	Fanny	Barbara	Total
Before 1	?	6 u	80
Change 1	+ 1 u	- 1 u
After 1	50	5 u	80
Before 2	5 p	5 u	80
Change 2	- 1 p	+ 1 p
After 2	4 p (40)	40	80

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

Number of buttons that Barbara and Fanny each had in the end is the same.

Number of buttons that Fanny had in the end
= 80 ÷ 2
= 40

Number of buttons that Fanny had in the end = 4 p

4 p = 40

1 p = 40 ÷ 4 = 10

Number of buttons that Fanny had after receiving some buttons from Barbara
= 5 p
= 5 x 10
= 50

Number of buttons that Barbara had after giving to Fanny
= 80 - 50
= 30

5 u = 30

1 u = 30 ÷ 5 = 6

Number of buttons that Barbara had at first
= 6 u
= 6 x 6
= 36

Number of buttons that Fanny had at first
= 80 - 36
= 44

Answer(s): 44