Natalie and Gwen had a total of 30 coins. Gwen gave ¹₄ of her coins to Natalie. In return, Natalie gave ¹₆ of the total number of coins that she had to Gwen. In the end, each girl had the same number of coins. How many coins did Natalie have at first?

	Natalie	Gwen	Total
Before 1	?	4 u	30
Change 1	+ 1 u	- 1 u
After 1	18	3 u	30
Before 2	6 p	3 u	30
Change 2	- 1 p	+ 1 p
After 2	5 p (15)	15	30

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

Number of coins that Gwen and Natalie each had in the end is the same.

Number of coins that Natalie had in the end
= 30 ÷ 2
= 15

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

5 p = 15

1 p = 15 ÷ 5 = 3

Number of coins that Natalie had after receiving some coins from Gwen
= 6 p
= 6 x 3
= 18

Number of coins that Gwen had after giving to Natalie
= 30 - 18
= 12

3 u = 12

1 u = 12 ÷ 3 = 4

Number of coins that Gwen had at first
= 4 u
= 4 x 4
= 16

Number of coins that Natalie had at first
= 30 - 16
= 14

Answer(s): 14