Opal and Jane had a total of 60 buttons. Jane gave ¹₅ of her buttons to Opal. In return, Opal gave ¹₄ of the total number of buttons that she had to Jane. In the end, each girl had the same number of buttons. How many buttons did Opal have at first?

	Opal	Jane	Total
Before 1	?	5 u	60
Change 1	+ 1 u	- 1 u
After 1	40	4 u	60
Before 2	4 p	4 u	60
Change 2	- 1 p	+ 1 p
After 2	3 p (30)	30	60

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

Number of buttons that Jane and Opal each had in the end is the same.

Number of buttons that Opal had in the end
= 60 ÷ 2
= 30

Number of buttons that Opal had in the end = 3 p

3 p = 30

1 p = 30 ÷ 3 = 10

Number of buttons that Opal had after receiving some buttons from Jane
= 4 p
= 4 x 10
= 40

Number of buttons that Jane had after giving to Opal
= 60 - 40
= 20

4 u = 20

1 u = 20 ÷ 4 = 5

Number of buttons that Jane had at first
= 5 u
= 5 x 5
= 25

Number of buttons that Opal had at first
= 60 - 25
= 35

Answer(s): 35