Ian had ³₄ as many cards as Howard. Howard had ⁴₇ as many cards as Cindy. After Ian and Cindy gave 24 cards to Howard, Howard had thrice as many cards as Ian. Cindy then had the same number of cards as Howard.

1. What was the ratio of the number of cards Ian had to the number of cards Cindy had in the beginning? Answer in simplest form.
2. How many cards did Howard have in the end?

Ian	Howard	Cindy
3	4
	4	7
3	4	7

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

Ratio of the number of cards that Ian had to the number of cards that Cindy had in the beginning = 3 : 7

	Ian	Howard	Cindy	Total
Before	3x1 = 3 u	4x1 = 4 u	7x1 = 7 u	14x1 = 14 u
Change	- 24	+ 48	- 24
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 Howard received from Ian and Cindy
= 2 x 24
= 48

Number of cards that Howard received from Ian and Cindy
= 2 x 1 u
= 2 u

2 u = 24

1 u = 24 ÷ 2 = 12

Number of cards that Ashley had in the end
= 6 u
= 6 x 12
= 72