Harper had 50% more cards than Emma. Joseph had 20% fewer cards than Harper. Harper and Emma gave Joseph some cards in the ratio of 4 : 5. As a result, Joseph had 50% more cards than before. Given that Joseph had 408 more cards than Emma in the end, how many cards did Harper give to Joseph?

Harper	Emma	Joseph
3x5	2x5
5x3		4x3
15	10	12

100% + 50% = 150%
150% = ¹⁵⁰₁₀₀ = ³₂

Harper : Emma = 3 : 2

100% - 20% = 80%
80% = ⁸⁰₁₀₀ = ⁴₅

Harper : Joseph = 5 : 4

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

	Harper	Emma	Joseph
Before	15x3 =  45 u	10x3 = 30 u	12x3 = 36 u
Change			+ 6x3 = + 18 u
Comparing the changes in the number of cards	- 4x2 = - 8 u	- 5x2 = - 10 u	+ 9x2 = + 18 u
After	37 u	20 u	18x3 = 54 u

50% x 12 
= ⁵⁰₁₀₀ x 12 
= 6 

The total number of cards that Harper and Emma gave to Joseph is repeated.  Make the total number of cards that Harper and Emma gave to Joseph the same.  LCM of 9 and 6 is 18.

Number of cards that Joseph had more than Emma 
= 54 u - 20 u
= 34 u

34 u = 408

1 u = 408 ÷ 34 = 12

Number of cards that Harper gave to Joseph
= 8 u
= 8 x 12
= 96

Answer(s): 96