Victoria and Lucy had a total of 70 buttons. Lucy gave ¹₃ of her buttons to Victoria. In return, Victoria gave ¹₆ of the total number of buttons that she had to Lucy. In the end, each girl had the same number of buttons. How many buttons did Victoria have at first?

	Victoria	Lucy	Total
Before 1	?	3 u	70
Change 1	+ 1 u	- 1 u
After 1	42	2 u	70
Before 2	6 p	2 u	70
Change 2	- 1 p	+ 1 p
After 2	5 p (35)	35	70

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

Number of buttons that Lucy and Victoria each had in the end is the same.

Number of buttons that Victoria had in the end
= 70 ÷ 2
= 35

Number of buttons that Victoria had in the end = 5 p

5 p = 35

1 p = 35 ÷ 5 = 7

Number of buttons that Victoria had after receiving some buttons from Lucy
= 6 p
= 6 x 7
= 42

Number of buttons that Lucy had after giving to Victoria
= 70 - 42
= 28

2 u = 28

1 u = 28 ÷ 2 = 14

Number of buttons that Lucy had at first
= 3 u
= 3 x 14
= 42

Number of buttons that Victoria had at first
= 70 - 42
= 28

Answer(s): 28