Gwen and Gabby had a total of 126 marbles. Gabby gave ¹₄ of her marbles to Gwen. In return, Gwen gave ¹₈ of the total number of marbles that she had to Gabby. In the end, each girl had the same number of marbles. How many marbles did Gwen have at first?

	Gwen	Gabby	Total
Before 1	?	4 u	126
Change 1	+ 1 u	- 1 u
After 1	72	3 u	126
Before 2	8 p	3 u	126
Change 2	- 1 p	+ 1 p
After 2	7 p (63)	63	126

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

Number of marbles that Gabby and Gwen each had in the end is the same.

Number of marbles that Gwen had in the end
= 126 ÷ 2
= 63

Number of marbles that Gwen had in the end = 7 p

7 p = 63

1 p = 63 ÷ 7 = 9

Number of marbles that Gwen had after receiving some marbles from Gabby
= 8 p
= 8 x 9
= 72

Number of marbles that Gabby had after giving to Gwen
= 126 - 72
= 54

3 u = 54

1 u = 54 ÷ 3 = 18

Number of marbles that Gabby had at first
= 4 u
= 4 x 18
= 72

Number of marbles that Gwen had at first
= 126 - 72
= 54

Answer(s): 54