Emma had 40% more cards than Ella. Hannah had 20% fewer cards than Emma. Emma and Ella gave Hannah some cards in the ratio of 3 : 4. As a result, Hannah had 50% more cards than before. Given that Hannah had 300 more cards than Ella in the end, how many cards did Emma give to Hannah?

Emma	Ella	Hannah
7x5	5x5
5x7		4x7
35	25	28

100% + 40% = 140%
140% = ¹⁴⁰₁₀₀ = ⁷₅

Emma : Ella = 7 : 5

100% - 20% = 80%
80% = ⁸⁰₁₀₀ = ⁴₅

Emma : Hannah = 5 : 4

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

	Emma	Ella	Hannah
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 Emma and Ella gave to Hannah is repeated.  Make the total number of cards that Emma and Ella gave to Hannah the same.  LCM of 7 and 14 is 14.

Number of cards that Hannah had more than Ella 
= 42 u - 17 u
= 25 u

25 u = 300

1 u = 300 ÷ 25 = 12

Number of cards that Emma gave to Hannah
= 6 u
= 6 x 12
= 72

Answer(s): 72