Bryan had ³₄ as many buttons as Julian. Julian had ⁴₇ as many buttons as Joelle. After Bryan and Joelle gave 12 buttons to Julian, Julian had thrice as many buttons as Bryan. Joelle then had the same number of buttons as Julian.

1. What was the ratio of the number of buttons Bryan had to the number of buttons Joelle had in the beginning? Answer in simplest form.
2. How many buttons did Julian have in the end?

Bryan	Julian	Joelle
3	4
	4	7
3	4	7

(a)
The number of buttons that Julian is repeated.  

Ratio of the number of buttons that Bryan had to the number of buttons that Joelle had in the beginning = 3 : 7

	Bryan	Julian	Joelle	Total
Before	3x1 = 3 u	4x1 = 4 u	7x1 = 7 u	14x1 = 14 u
Change	- 12	+ 24	- 12
After	1x2 = 2 u	3x2 = 6 u	3x2 = 6 u	7x2 = 14 u

The total number of buttons at first and in the end remains unchanged. Make the total number of buttons the same. LCM of 14 and 7 is 14.

Number of buttons that Julian received from Bryan and Joelle
= 2 x 12
= 24

Number of buttons that Julian received from Bryan and Joelle
= 2 x 1 u
= 2 u

2 u = 12

1 u = 12 ÷ 2 = 6

Number of buttons that Ashley had in the end
= 6 u
= 6 x 6
= 36