Nick had ³₄ as many buttons as Jack. Jack had ⁴₇ as many buttons as Victoria. After Nick and Victoria gave 22 buttons to Jack, Jack had thrice as many buttons as Nick. Victoria then had the same number of buttons as Jack.

1. What was the ratio of the number of buttons Victoria had to the number of buttons Nick had in the beginning? Answer in simplest form.
2. How many buttons did Jack have in the end?

Nick	Jack	Victoria
3	4
	4	7
3	4	7

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

Ratio of the number of buttons that Victoria had to the number of buttons that Nick had in the beginning = 7 : 3

	Nick	Jack	Victoria	Total
Before	3x1 = 3 u	4x1 = 4 u	7x1 = 7 u	14x1 = 14 u
Change	- 22	+ 44	- 22
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 Jack received from Nick and Victoria
= 2 x 22
= 44

Number of buttons that Jack received from Nick and Victoria
= 2 x 1 u
= 2 u

2 u = 22

1 u = 22 ÷ 2 = 11

Number of buttons that Ashley had in the end
= 6 u
= 6 x 11
= 66