Luke had ³₄ as many buttons as Oliver. Oliver had ⁴₇ as many buttons as Jaslyn. After Luke and Jaslyn gave 28 buttons to Oliver, Oliver had thrice as many buttons as Luke. Jaslyn then had the same number of buttons as Oliver.

1. What was the ratio of the number of buttons Luke had to the number of buttons Jaslyn had in the beginning? Answer in simplest form.
2. How many buttons did Oliver have in the end?

Luke	Oliver	Jaslyn
3	4
	4	7
3	4	7

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

Ratio of the number of buttons that Luke had to the number of buttons that Jaslyn had in the beginning = 3 : 7

	Luke	Oliver	Jaslyn	Total
Before	3x1 = 3 u	4x1 = 4 u	7x1 = 7 u	14x1 = 14 u
Change	- 28	+ 56	- 28
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 Oliver received from Luke and Jaslyn
= 2 x 28
= 56

Number of buttons that Oliver received from Luke and Jaslyn
= 2 x 1 u
= 2 u

2 u = 28

1 u = 28 ÷ 2 = 14

Number of buttons that Ashley had in the end
= 6 u
= 6 x 14
= 84