Kathy and Ivory had a total of 50 coins. Ivory gave ¹₃ of her coins to Kathy. In return, Kathy gave ¹₆ of the total number of coins that she had to Ivory. In the end, each girl had the same number of coins. How many coins did Kathy have at first?

	Kathy	Ivory	Total
Before 1	?	3 u	50
Change 1	+ 1 u	- 1 u
After 1	30	2 u	50
Before 2	6 p	2 u	50
Change 2	- 1 p	+ 1 p
After 2	5 p (25)	25	50

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

Number of coins that Ivory and Kathy each had in the end is the same.

Number of coins that Kathy had in the end
= 50 ÷ 2
= 25

Number of coins that Kathy had in the end = 5 p

5 p = 25

1 p = 25 ÷ 5 = 5

Number of coins that Kathy had after receiving some coins from Ivory
= 6 p
= 6 x 5
= 30

Number of coins that Ivory had after giving to Kathy
= 50 - 30
= 20

2 u = 20

1 u = 20 ÷ 2 = 10

Number of coins that Ivory had at first
= 3 u
= 3 x 10
= 30

Number of coins that Kathy had at first
= 50 - 30
= 20

Answer(s): 20