Esther and Gillian had a total of 48 marbles. Gillian gave ¹₅ of her marbles to Esther. In return, Esther gave ¹₇ of the total number of marbles that she had to Gillian. In the end, each girl had the same number of marbles. How many marbles did Esther have at first?

	Esther	Gillian	Total
Before 1	?	5 u	48
Change 1	+ 1 u	- 1 u
After 1	28	4 u	48
Before 2	7 p	4 u	48
Change 2	- 1 p	+ 1 p
After 2	6 p (24)	24	48

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

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

Number of marbles that Esther had in the end
= 48 ÷ 2
= 24

Number of marbles that Esther had in the end = 6 p

6 p = 24

1 p = 24 ÷ 6 = 4

Number of marbles that Esther had after receiving some marbles from Gillian
= 7 p
= 7 x 4
= 28

Number of marbles that Gillian had after giving to Esther
= 48 - 28
= 20

4 u = 20

1 u = 20 ÷ 4 = 5

Number of marbles that Gillian had at first
= 5 u
= 5 x 5
= 25

Number of marbles that Esther had at first
= 48 - 25
= 23

Answer(s): 23