Henry had 40% more cards than Zoey. Raeann had 20% fewer cards than Henry. Henry and Zoey gave Raeann some cards in the ratio of 3 : 4. As a result, Raeann had 50% more cards than before. Given that Raeann had 225 more cards than Zoey in the end, how many cards did Henry give to Raeann?

Henry	Zoey	Raeann
7x5	5x5
5x7		4x7
35	25	28

100% + 40% = 140%
140% = ¹⁴⁰₁₀₀ = ⁷₅

Henry : Zoey = 7 : 5

100% - 20% = 80%
80% = ⁸⁰₁₀₀ = ⁴₅

Henry : Raeann = 5 : 4

The number of cards that Henry had at first is repeated. Make the number of cards that Henry had at first the same.  LCM of 7 and 5 is 35.

	Henry	Zoey	Raeann
Before	35x1 =  35 u	25x1 = 25 u	28x1 = 28 u
Change			+ 14x1 = + 14 u
Comparing the changes in the number of cards	- 3x2 = - 6 u	- 4x2 = - 8 u	+ 7x2 = + 14 u
After	29 u	17 u	42x1 = 42 u

50% x 28 
= ⁵⁰₁₀₀ x 28 
= 14 

The total number of cards that Henry and Zoey gave to Raeann is repeated.  Make the total number of cards that Henry and Zoey gave to Raeann the same.  LCM of 7 and 14 is 14.

Number of cards that Raeann had more than Zoey 
= 42 u - 17 u
= 25 u

25 u = 225

1 u = 225 ÷ 25 = 9

Number of cards that Henry gave to Raeann
= 6 u
= 6 x 9
= 54

Answer(s): 54