Michael had 50% more cards than Rachel. Sophia had 20% fewer cards than Michael. Michael and Rachel gave Sophia some cards in the ratio of 3 : 5. As a result, Sophia had 50% more cards than before. Given that Sophia had 423 more cards than Rachel in the end, how many cards did Michael give to Sophia?

Michael	Rachel	Sophia
3x5	2x5
5x3		4x3
15	10	12

100% + 50% = 150%
150% = ¹⁵⁰₁₀₀ = ³₂

Michael : Rachel = 3 : 2

100% - 20% = 80%
80% = ⁸⁰₁₀₀ = ⁴₅

Michael : Sophia = 5 : 4

The number of cards that Michael had at first is repeated. Make the number of cards that Michael had at first the same.  LCM of 3 and 5 is 15.

	Michael	Rachel	Sophia
Before	15x4 =  60 u	10x4 = 40 u	12x4 = 48 u
Change			+ 6x4 = + 24 u
Comparing the changes in the number of cards	- 3x3 = - 9 u	- 5x3 = - 15 u	+ 8x3 = + 24 u
After	51 u	25 u	18x4 = 72 u

50% x 12 
= ⁵⁰₁₀₀ x 12 
= 6 

The total number of cards that Michael and Rachel gave to Sophia is repeated.  Make the total number of cards that Michael and Rachel gave to Sophia the same.  LCM of 8 and 6 is 24.

Number of cards that Sophia had more than Rachel 
= 72 u - 25 u
= 47 u

47 u = 423

1 u = 423 ÷ 47 = 9

Number of cards that Michael gave to Sophia
= 9 u
= 9 x 9
= 81

Answer(s): 81