Valen, Seth and Zeph had some buttons in the ratio of 6 : 7 : 13. Zeph gave 50% of his buttons to Valen and Seth. As a result, Valen had the same number of buttons as Seth and Zeph had 39 fewer buttons than Valen.

1. What was the percentage increase in Valen's buttons after receiving from Zeph?
2. How many buttons did Seth have at first?

Valen	Seth	Zeph	Total
6 u	7 u	13 u	26 u
+ 6.5 u		- 6.5 u
9.75 u	9.75 u	6.5 u	26 u

(a)
Number of buttons that Zeph gave away
= 50% x 13
= ⁵⁰₁₀₀ x 13
= 6.5 u

Number of buttons that Valen and Seth each had in the end
= (26 u - 6.5 u) ÷ 2
= 19.5 u ÷ 2
= 9.75 u

Increase in the number of buttons that Valen had
= 9.75 u - 6 u
= 3.75 u

Percentage increase in Valen's buttons after receiving from Zeph
= ^3.75₆ x 100%
= 62.5%

(b)
9.75 u = 6.5 u + 39
9.75 u - 6.5 u = 39
3.25 u = 39

1 u = 39 ÷ 3.25 = 12

Number of buttons that Seth had at first
= 7 u
= 7 x 12
= 84

Answer(s): (a) 62.5%; (b) 84