Gem and Ivory had a total of 36 pens. Ivory gave ¹₄ of her pens to Gem. In return, Gem gave ¹₃ of the total number of pens that she had to Ivory. In the end, each girl had the same number of pens. How many pens did Gem have at first?

	Gem	Ivory	Total
Before 1	?	4 u	36
Change 1	+ 1 u	- 1 u
After 1	27	3 u	36
Before 2	3 p	3 u	36
Change 2	- 1 p	+ 1 p
After 2	2 p (18)	18	36

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

Number of pens that Ivory and Gem each had in the end is the same.

Number of pens that Gem had in the end
= 36 ÷ 2
= 18

Number of pens that Gem had in the end = 2 p

2 p = 18

1 p = 18 ÷ 2 = 9

Number of pens that Gem had after receiving some pens from Ivory
= 3 p
= 3 x 9
= 27

Number of pens that Ivory had after giving to Gem
= 36 - 27
= 9

3 u = 9

1 u = 9 ÷ 3 = 3

Number of pens that Ivory had at first
= 4 u
= 4 x 3
= 12

Number of pens that Gem had at first
= 36 - 12
= 24

Answer(s): 24