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

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

Number of beads that Gwen and Nora each had in the end is the same.

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

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

2 p = 24

1 p = 24 ÷ 2 = 12

Number of beads that Nora had after receiving some beads from Gwen
= 3 p
= 3 x 12
= 36

Number of beads that Gwen had after giving to Nora
= 48 - 36
= 12

4 u = 12

1 u = 12 ÷ 4 = 3

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

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

Answer(s): 33