Luke had ³₄ as many marbles as Flynn. Flynn had ⁴₇ as many marbles as Penelope. After Luke and Penelope gave 14 marbles to Flynn, Flynn had thrice as many marbles as Luke. Penelope then had the same number of marbles as Flynn.

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

Luke	Flynn	Penelope
3	4
	4	7
3	4	7

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

Ratio of the number of marbles that Luke had to the number of marbles that Penelope had in the beginning = 3 : 7

	Luke	Flynn	Penelope	Total
Before	3x1 = 3 u	4x1 = 4 u	7x1 = 7 u	14x1 = 14 u
Change	- 14	+ 28	- 14
After	1x2 = 2 u	3x2 = 6 u	3x2 = 6 u	7x2 = 14 u

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

Number of marbles that Flynn received from Luke and Penelope
= 2 x 14
= 28

Number of marbles that Flynn received from Luke and Penelope
= 2 x 1 u
= 2 u

2 u = 14

1 u = 14 ÷ 2 = 7

Number of marbles that Ashley had in the end
= 6 u
= 6 x 7
= 42