Gem and Eva had a total of 90 erasers. Eva gave ¹₄ of her erasers to Gem. In return, Gem gave ¹₆ of the total number of erasers that she had to Eva. In the end, each girl had the same number of erasers. How many erasers did Gem have at first?

	Gem	Eva	Total
Before 1	?	4 u	90
Change 1	+ 1 u	- 1 u
After 1	54	3 u	90
Before 2	6 p	3 u	90
Change 2	- 1 p	+ 1 p
After 2	5 p (45)	45	90

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

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

Number of erasers that Gem had in the end
= 90 ÷ 2
= 45

Number of erasers that Gem had in the end = 5 p

5 p = 45

1 p = 45 ÷ 5 = 9

Number of erasers that Gem had after receiving some erasers from Eva
= 6 p
= 6 x 9
= 54

Number of erasers that Eva had after giving to Gem
= 90 - 54
= 36

3 u = 36

1 u = 36 ÷ 3 = 12

Number of erasers that Eva had at first
= 4 u
= 4 x 12
= 48

Number of erasers that Gem had at first
= 90 - 48
= 42

Answer(s): 42