Elyse and Gabby had a total of 120 coins. Gabby gave ¹₃ of her coins to Elyse. In return, Elyse gave ¹₆ of the total number of coins that she had to Gabby. In the end, each girl had the same number of coins. How many coins did Elyse have at first?

	Elyse	Gabby	Total
Before 1	?	3 u	120
Change 1	+ 1 u	- 1 u
After 1	72	2 u	120
Before 2	6 p	2 u	120
Change 2	- 1 p	+ 1 p
After 2	5 p (60)	60	120

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

Number of coins that Gabby and Elyse each had in the end is the same.

Number of coins that Elyse had in the end
= 120 ÷ 2
= 60

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

5 p = 60

1 p = 60 ÷ 5 = 12

Number of coins that Elyse had after receiving some coins from Gabby
= 6 p
= 6 x 12
= 72

Number of coins that Gabby had after giving to Elyse
= 120 - 72
= 48

2 u = 48

1 u = 48 ÷ 2 = 24

Number of coins that Gabby had at first
= 3 u
= 3 x 24
= 72

Number of coins that Elyse had at first
= 120 - 72
= 48

Answer(s): 48