Michael had ³₄ as many cards as Luke. Luke had ⁴₇ as many cards as Julie. After Michael and Julie gave 10 cards to Luke, Luke had thrice as many cards as Michael. Julie then had the same number of cards as Luke.

1. What was the ratio of the number of cards Michael had to the number of cards Julie had in the beginning? Answer in simplest form.
2. How many cards did Luke have in the end?

Michael	Luke	Julie
3	4
	4	7
3	4	7

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

Ratio of the number of cards that Michael had to the number of cards that Julie had in the beginning = 3 : 7

	Michael	Luke	Julie	Total
Before	3x1 = 3 u	4x1 = 4 u	7x1 = 7 u	14x1 = 14 u
Change	- 10	+ 20	- 10
After	1x2 = 2 u	3x2 = 6 u	3x2 = 6 u	7x2 = 14 u

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

Number of cards that Luke received from Michael and Julie
= 2 x 10
= 20

Number of cards that Luke received from Michael and Julie
= 2 x 1 u
= 2 u

2 u = 10

1 u = 10 ÷ 2 = 5

Number of cards that Ashley had in the end
= 6 u
= 6 x 5
= 30