Barbara and Sarah had a total of 120 beads. Sarah gave ¹₃ of her beads to Barbara. In return, Barbara gave ¹₆ of the total number of beads that she had to Sarah. In the end, each girl had the same number of beads. How many beads did Barbara have at first?

	Barbara	Sarah	Total
Before 1	?	3 u	120
Change 1	+ 1 u	- 1 u
After 1	72	2 u	120
Before 2	6 p	2 u	120
Change 2	- 1 p	+ 1 p
After 2	5 p (60)	60	120

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

Number of beads that Sarah and Barbara each had in the end is the same.

Number of beads that Barbara had in the end
= 120 ÷ 2
= 60

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

5 p = 60

1 p = 60 ÷ 5 = 12

Number of beads that Barbara had after receiving some beads from Sarah
= 6 p
= 6 x 12
= 72

Number of beads that Sarah had after giving to Barbara
= 120 - 72
= 48

2 u = 48

1 u = 48 ÷ 2 = 24

Number of beads that Sarah had at first
= 3 u
= 3 x 24
= 72

Number of beads that Barbara had at first
= 120 - 72
= 48

Answer(s): 48