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

	Xuan	Olivia	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 Olivia gave some beads to Xuan and Xuan then gave some beads to Olivia, it is an internal transfer of beads between the two girls. So, the total number of beads remains unchanged.

Number of beads that Olivia and Xuan each had in the end is the same.

Number of beads that Xuan had in the end
= 70 ÷ 2
= 35

Number of beads that Xuan had in the end = 5 p

5 p = 35

1 p = 35 ÷ 5 = 7

Number of beads that Xuan had after receiving some beads from Olivia
= 6 p
= 6 x 7
= 42

Number of beads that Olivia had after giving to Xuan
= 70 - 42
= 28

2 u = 28

1 u = 28 ÷ 2 = 14

Number of beads that Olivia had at first
= 3 u
= 3 x 14
= 42

Number of beads that Xuan had at first
= 70 - 42
= 28

Answer(s): 28