Yen and Ivory had a total of 110 marbles. Ivory gave ¹₃ of her marbles to Yen. In return, Yen gave ¹₆ of the total number of marbles that she had to Ivory. In the end, each girl had the same number of marbles. How many marbles did Yen have at first?

	Yen	Ivory	Total
Before 1	?	3 u	110
Change 1	+ 1 u	- 1 u
After 1	66	2 u	110
Before 2	6 p	2 u	110
Change 2	- 1 p	+ 1 p
After 2	5 p (55)	55	110

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

Number of marbles that Ivory and Yen each had in the end is the same.

Number of marbles that Yen had in the end
= 110 ÷ 2
= 55

Number of marbles that Yen had in the end = 5 p

5 p = 55

1 p = 55 ÷ 5 = 11

Number of marbles that Yen had after receiving some marbles from Ivory
= 6 p
= 6 x 11
= 66

Number of marbles that Ivory had after giving to Yen
= 110 - 66
= 44

2 u = 44

1 u = 44 ÷ 2 = 22

Number of marbles that Ivory had at first
= 3 u
= 3 x 22
= 66

Number of marbles that Yen had at first
= 110 - 66
= 44

Answer(s): 44