Kathy and Tammy had a total of 98 buttons. Tammy gave ¹₄ of her buttons to Kathy. In return, Kathy gave ¹₈ of the total number of buttons that she had to Tammy. In the end, each girl had the same number of buttons. How many buttons did Kathy have at first?

	Kathy	Tammy	Total
Before 1	?	4 u	98
Change 1	+ 1 u	- 1 u
After 1	56	3 u	98
Before 2	8 p	3 u	98
Change 2	- 1 p	+ 1 p
After 2	7 p (49)	49	98

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

Number of buttons that Tammy and Kathy each had in the end is the same.

Number of buttons that Kathy had in the end
= 98 ÷ 2
= 49

Number of buttons that Kathy had in the end = 7 p

7 p = 49

1 p = 49 ÷ 7 = 7

Number of buttons that Kathy had after receiving some buttons from Tammy
= 8 p
= 8 x 7
= 56

Number of buttons that Tammy had after giving to Kathy
= 98 - 56
= 42

3 u = 42

1 u = 42 ÷ 3 = 14

Number of buttons that Tammy had at first
= 4 u
= 4 x 14
= 56

Number of buttons that Kathy had at first
= 98 - 56
= 42

Answer(s): 42