Ben had ³₄ as many marbles as Michael. Michael had ⁴₇ as many marbles as Lucy. After Ben and Lucy gave 4 marbles to Michael, Michael had thrice as many marbles as Ben. Lucy then had the same number of marbles as Michael.

1. What was the ratio of the number of marbles Lucy had to the number of marbles Ben had in the beginning? Answer in simplest form.
2. How many marbles did Michael have in the end?

Ben	Michael	Lucy
3	4
	4	7
3	4	7

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

Ratio of the number of marbles that Lucy had to the number of marbles that Ben had in the beginning = 7 : 3

	Ben	Michael	Lucy	Total
Before	3x1 = 3 u	4x1 = 4 u	7x1 = 7 u	14x1 = 14 u
Change	- 4	+ 8	- 4
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 Michael received from Ben and Lucy
= 2 x 4
= 8

Number of marbles that Michael received from Ben and Lucy
= 2 x 1 u
= 2 u

2 u = 4

1 u = 4 ÷ 2 = 2

Number of marbles that Ashley had in the end
= 6 u
= 6 x 2
= 12