Ethan had some buttons. He gave ⁵₆ of his buttons to Seth and bought 4 new buttons. After that, he gave ²₇ of his remaining buttons to his cousin and bought another 7 new buttons. In the end, Ethan had 27 buttons.

1. How many buttons did Ethan give to his cousin?
2. How many buttons did Ethan have at first?

(a)
Fraction of  his remaining buttons after he gave to his cousin
= 1 - ²₇
= ⁵₇

⁵₇ of his remaining buttons = 27 - 7 = 20
¹₇ of his remaining buttons = 20 ÷ 5 = 4
²₇ of his remaining buttons = 2 x 4 = 8

Number of buttons that Ethan gave to his cousin = 8

(b)
⁷₇ of his remaining buttons = 7 x 4 = 28

Fraction of his total buttons after he gave to Seth
= 1 - ⁵₆
= ¹₆

¹₆ of his total buttons = 28 - 4 = 24
⁶₆ of his total buttons = 6 x 24 = 144

Number of buttons that Ethan had at first = 144 

Answer(s): (a) 8; (b) 144