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

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

Number of marbles that Esther and Julie each had in the end is the same.

Number of marbles that Julie had in the end
= 112 ÷ 2
= 56

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

7 p = 56

1 p = 56 ÷ 7 = 8

Number of marbles that Julie had after receiving some marbles from Esther
= 8 p
= 8 x 8
= 64

Number of marbles that Esther had after giving to Julie
= 112 - 64
= 48

2 u = 48

1 u = 48 ÷ 2 = 24

Number of marbles that Esther had at first
= 3 u
= 3 x 24
= 72

Number of marbles that Julie had at first
= 112 - 72
= 40

Answer(s): 40