Gabriel had ¹₂ as many marbles as Luke. Luke had ²₃ as many marbles as Zara. After Gabriel and Zara gave 30 marbles to Luke, Luke had four times as many marbles as Gabriel. Zara then had the same number of marbles as Luke.

1. What was the ratio of the number of marbles Gabriel had to the number of marbles Zara had in the beginning? Answer in simplest form.
2. How many marbles did Luke have in the end?

Gabriel	Luke	Zara
1	2
	2	3
1	2	3

(a)
The number of marbles that Luke is repeated.  

Ratio of the number of marbles that Gabriel had to the number of marbles that Zara had in the beginning = 1 : 3

	Gabriel	Luke	Zara	Total
Before	1x3 = 3 u	2x3 = 6 u	3x3 = 9 u	6x3 = 18 u
Change	- 30	+ 60	- 30
After	1x2 = 2 u	4x2 = 8 u	4x2 = 8 u	9x2 = 18 u

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

Number of marbles that Luke received from Gabriel and Zara
= 2 x 30
= 60

Number of marbles that Luke received from Gabriel and Zara
= 2 x 1 u
= 2 u

2 u = 30

1 u = 30 ÷ 2 = 15

Number of marbles that Ashley had in the end
= 8 u
= 8 x 15
= 120