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

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

Number of erasers that Nora and Gem each had in the end is the same.

Number of erasers that Gem had in the end
= 112 ÷ 2
= 56

Number of erasers that Gem had in the end = 7 p

7 p = 56

1 p = 56 ÷ 7 = 8

Number of erasers that Gem had after receiving some erasers from Nora
= 8 p
= 8 x 8
= 64

Number of erasers that Nora had after giving to Gem
= 112 - 64
= 48

2 u = 48

1 u = 48 ÷ 2 = 24

Number of erasers that Nora had at first
= 3 u
= 3 x 24
= 72

Number of erasers that Gem had at first
= 112 - 72
= 40

Answer(s): 40