Peter had ¹₂ as many buttons as Sam. Sam had ²₃ as many buttons as Victoria. After Peter and Victoria gave 14 buttons to Sam, Sam had four times as many buttons as Peter. Victoria then had the same number of buttons as Sam.

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

Peter	Sam	Victoria
1	2
	2	3
1	2	3

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

Ratio of the number of buttons that Victoria had to the number of buttons that Peter had in the beginning = 3 : 1

	Peter	Sam	Victoria	Total
Before	1x3 = 3 u	2x3 = 6 u	3x3 = 9 u	6x3 = 18 u
Change	- 14	+ 28	- 14
After	1x2 = 2 u	4x2 = 8 u	4x2 = 8 u	9x2 = 18 u

The total number of buttons at first and in the end remains unchanged. Make the total number of buttons the same. LCM of 6 and 9 is 18.

Number of buttons that Sam received from Peter and Victoria
= 2 x 14
= 28

Number of buttons that Sam received from Peter and Victoria
= 2 x 1 u
= 2 u

2 u = 14

1 u = 14 ÷ 2 = 7

Number of buttons that Ashley had in the end
= 8 u
= 8 x 7
= 56