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

	Joelle	Zoe	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 Zoe gave some buttons to Joelle and Joelle then gave some buttons to Zoe, it is an internal transfer of buttons between the two girls. So, the total number of buttons remains unchanged.

Number of buttons that Zoe and Joelle each had in the end is the same.

Number of buttons that Joelle had in the end
= 48 ÷ 2
= 24

Number of buttons that Joelle had in the end = 4 p

4 p = 24

1 p = 24 ÷ 4 = 6

Number of buttons that Joelle had after receiving some buttons from Zoe
= 5 p
= 5 x 6
= 30

Number of buttons that Zoe had after giving to Joelle
= 48 - 30
= 18

2 u = 18

1 u = 18 ÷ 2 = 9

Number of buttons that Zoe had at first
= 3 u
= 3 x 9
= 27

Number of buttons that Joelle had at first
= 48 - 27
= 21

Answer(s): 21