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

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

Number of beads that Penelope and Roshel each had in the end is the same.

Number of beads that Roshel had in the end
= 80 ÷ 2
= 40

Number of beads that Roshel had in the end = 4 p

4 p = 40

1 p = 40 ÷ 4 = 10

Number of beads that Roshel had after receiving some beads from Penelope
= 5 p
= 5 x 10
= 50

Number of beads that Penelope had after giving to Roshel
= 80 - 50
= 30

5 u = 30

1 u = 30 ÷ 5 = 6

Number of beads that Penelope had at first
= 6 u
= 6 x 6
= 36

Number of beads that Roshel had at first
= 80 - 36
= 44

Answer(s): 44