Sarah and Olivia had a total of 40 beads. Olivia gave ¹₃ of her beads to Sarah. In return, Sarah 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 Sarah have at first?

	Sarah	Olivia	Total
Before 1	?	3 u	40
Change 1	+ 1 u	- 1 u
After 1	24	2 u	40
Before 2	6 p	2 u	40
Change 2	- 1 p	+ 1 p
After 2	5 p (20)	20	40

Since Olivia gave some beads to Sarah and Sarah 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 Sarah each had in the end is the same.

Number of beads that Sarah had in the end
= 40 ÷ 2
= 20

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

5 p = 20

1 p = 20 ÷ 5 = 4

Number of beads that Sarah had after receiving some beads from Olivia
= 6 p
= 6 x 4
= 24

Number of beads that Olivia had after giving to Sarah
= 40 - 24
= 16

2 u = 16

1 u = 16 ÷ 2 = 8

Number of beads that Olivia had at first
= 3 u
= 3 x 8
= 24

Number of beads that Sarah had at first
= 40 - 24
= 16

Answer(s): 16