Liam, Ryan and Japheth had some buttons in the ratio of 6 : 7 : 13. Japheth gave 50% of his buttons to Liam and Ryan. As a result, Liam had the same number of buttons as Ryan and Japheth had 13 fewer buttons than Liam.

1. What was the percentage increase in Liam's buttons after receiving from Japheth?
2. How many buttons did Ryan have at first?

Liam	Ryan	Japheth	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 Japheth gave away
= 50% x 13
= ⁵⁰₁₀₀ x 13
= 6.5 u

Number of buttons that Liam and Ryan 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 Liam had
= 9.75 u - 6 u
= 3.75 u

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

(b)
9.75 u = 6.5 u + 13
9.75 u - 6.5 u = 13
3.25 u = 13

1 u = 13 ÷ 3.25 = 4

Number of buttons that Ryan had at first
= 7 u
= 7 x 4
= 28

Answer(s): (a) 62.5%; (b) 28