Hilda and Hazel had a total of 112 beads. Hazel gave ¹₃ of her beads to Hilda. In return, Hilda gave ¹₈ of the total number of beads that she had to Hazel. In the end, each girl had the same number of beads. How many beads did Hilda have at first?

	Hilda	Hazel	Total
Before 1	?	3 u	112
Change 1	+ 1 u	- 1 u
After 1	64	2 u	112
Before 2	8 p	2 u	112
Change 2	- 1 p	+ 1 p
After 2	7 p (56)	56	112

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

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

Number of beads that Hilda had in the end
= 112 ÷ 2
= 56

Number of beads that Hilda had in the end = 7 p

7 p = 56

1 p = 56 ÷ 7 = 8

Number of beads that Hilda had after receiving some beads from Hazel
= 8 p
= 8 x 8
= 64

Number of beads that Hazel had after giving to Hilda
= 112 - 64
= 48

2 u = 48

1 u = 48 ÷ 2 = 24

Number of beads that Hazel had at first
= 3 u
= 3 x 24
= 72

Number of beads that Hilda had at first
= 112 - 72
= 40

Answer(s): 40