Penelope and Yen had a total of 48 beads. Yen gave ¹₅ of her beads to Penelope. In return, Penelope gave ¹₃ of the total number of beads that she had to Yen. In the end, each girl had the same number of beads. How many beads did Penelope have at first?

	Penelope	Yen	Total
Before 1	?	5 u	48
Change 1	+ 1 u	- 1 u
After 1	36	4 u	48
Before 2	3 p	4 u	48
Change 2	- 1 p	+ 1 p
After 2	2 p (24)	24	48

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

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

Number of beads that Penelope had in the end
= 48 ÷ 2
= 24

Number of beads that Penelope had in the end = 2 p

2 p = 24

1 p = 24 ÷ 2 = 12

Number of beads that Penelope had after receiving some beads from Yen
= 3 p
= 3 x 12
= 36

Number of beads that Yen had after giving to Penelope
= 48 - 36
= 12

4 u = 12

1 u = 12 ÷ 4 = 3

Number of beads that Yen had at first
= 5 u
= 5 x 3
= 15

Number of beads that Penelope had at first
= 48 - 15
= 33

Answer(s): 33