Esther and Irene had a total of 48 marbles. Irene gave ¹₃ of her marbles to Esther. In return, Esther gave ¹₅ of the total number of marbles that she had to Irene. In the end, each girl had the same number of marbles. How many marbles did Esther have at first?

	Esther	Irene	Total
Before 1	?	3 u	48
Change 1	+ 1 u	- 1 u
After 1	30	2 u	48
Before 2	5 p	2 u	48
Change 2	- 1 p	+ 1 p
After 2	4 p (24)	24	48

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

Number of marbles that Irene 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 = 4 p

4 p = 24

1 p = 24 ÷ 4 = 6

Number of marbles that Esther had after receiving some marbles from Irene
= 5 p
= 5 x 6
= 30

Number of marbles that Irene had after giving to Esther
= 48 - 30
= 18

2 u = 18

1 u = 18 ÷ 2 = 9

Number of marbles that Irene had at first
= 3 u
= 3 x 9
= 27

Number of marbles that Esther had at first
= 48 - 27
= 21

Answer(s): 21