Rael had 25% more pencils than Josiah. Hannah had 50% fewer pencils than Rael. Rael and Josiah gave Hannah some pencils in the ratio of 1 : 3. As a result, Hannah had 60% more pencils than before. Given that Josiah had 54 less pencils than Hannah in the end, how many pencils did Josiah give to Hannah?

Rael	Josiah	Hannah
5x2	4x2
2x5		1x5
10	8	5

100% + 25% = 125%
125% = ¹²⁵₁₀₀ = ⁵₄

Rael : Josiah = 5 : 4

100% - 50% = 50%
50% = ⁵⁰₁₀₀ = ¹₂

Rael : Hannah = 2 : 1

The number of pencils that Rael had at first is repeated. Make the number of pencils that Rael had at first the same.  LCM of 5 and 2 is 10.

	Rael	Josiah	Hannah
Before	10x4 =  40 u	8x4 = 32 u	5x4 = 20 u
Change			+ 3x4 = + 12 u
Comparing the changes in the number of pencils	- 1x3 = - 3 u	- 3x3 = - 9 u	+ 4x3 = + 12 u
After	37 u	23 u	8x4 = 32 u

60% x 5 
= ⁶⁰₁₀₀ x 5 
= 3 

The total number of pencils that Rael and Josiah gave to Hannah is repeated.  Make the total number of pencils that Rael and Josiah gave to Hannah the same.  LCM of 4 and 3 is 12.

Number of pencils that Josiah had less than Hannah 
= 32 u - 23 u
= 9 u

9 u = 54

1 u = 54 ÷ 9 = 6

Number of pencils that Josiah gave to Hannah
= 9 u
= 9 x 6
= 54

Answer(s): 54