Ella had 40% more cards than Ethan. Jack had 20% fewer cards than Ella. Ella and Ethan gave Jack some cards in the ratio of 1 : 3. As a result, Jack had 50% more cards than before. Given that Jack had 275 more cards than Ethan in the end, how many cards did Ella give to Jack?

Ella	Ethan	Jack
7x5	5x5
5x7		4x7
35	25	28

100% + 40% = 140%
140% = ¹⁴⁰₁₀₀ = ⁷₅

Ella : Ethan = 7 : 5

100% - 20% = 80%
80% = ⁸⁰₁₀₀ = ⁴₅

Ella : Jack = 5 : 4

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

	Ella	Ethan	Jack
Before	35x2 =  70 u	25x2 = 50 u	28x2 = 56 u
Change			+ 14x2 = + 28 u
Comparing the changes in the number of cards	- 1x7 = - 7 u	- 3x7 = - 21 u	+ 4x7 = + 28 u
After	63 u	29 u	42x2 = 84 u

50% x 28 
= ⁵⁰₁₀₀ x 28 
= 14 

The total number of cards that Ella and Ethan gave to Jack is repeated.  Make the total number of cards that Ella and Ethan gave to Jack the same.  LCM of 4 and 14 is 28.

Number of cards that Jack had more than Ethan 
= 84 u - 29 u
= 55 u

55 u = 275

1 u = 275 ÷ 55 = 5

Number of cards that Ella gave to Jack
= 7 u
= 7 x 5
= 35

Answer(s): 35