Hazel and Gabby had a total of 48 beads. Gabby gave ¹₄ of her beads to Hazel. In return, Hazel gave ¹₃ of the total number of beads that she had to Gabby. In the end, each girl had the same number of beads. How many beads did Hazel have at first?

	Hazel	Gabby	Total
Before 1	?	4 u	48
Change 1	+ 1 u	- 1 u
After 1	36	3 u	48
Before 2	3 p	3 u	48
Change 2	- 1 p	+ 1 p
After 2	2 p (24)	24	48

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

Number of beads that Gabby and Hazel each had in the end is the same.

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

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

2 p = 24

1 p = 24 ÷ 2 = 12

Number of beads that Hazel had after receiving some beads from Gabby
= 3 p
= 3 x 12
= 36

Number of beads that Gabby had after giving to Hazel
= 48 - 36
= 12

3 u = 12

1 u = 12 ÷ 3 = 4

Number of beads that Gabby had at first
= 4 u
= 4 x 4
= 16

Number of beads that Hazel had at first
= 48 - 16
= 32

Answer(s): 32