Josiah had 50% more pens than Owen. Hannah had 20% fewer pens than Josiah. Josiah and Owen gave Hannah some pens in the ratio of 1 : 3. As a result, Hannah had 50% more pens than before. Given that Hannah had 150 more pens than Owen in the end, how many pens did Josiah give to Hannah?

Josiah	Owen	Hannah
3x5	2x5
5x3		4x3
15	10	12

100% + 50% = 150%
150% = ¹⁵⁰₁₀₀ = ³₂

Josiah : Owen = 3 : 2

100% - 20% = 80%
80% = ⁸⁰₁₀₀ = ⁴₅

Josiah : Hannah = 5 : 4

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

	Josiah	Owen	Hannah
Before	15x2 =  30 u	10x2 = 20 u	12x2 = 24 u
Change			+ 6x2 = + 12 u
Comparing the changes in the number of pens	- 1x3 = - 3 u	- 3x3 = - 9 u	+ 4x3 = + 12 u
After	27 u	11 u	18x2 = 36 u

50% x 12 
= ⁵⁰₁₀₀ x 12 
= 6 

The total number of pens that Josiah and Owen gave to Hannah is repeated.  Make the total number of pens that Josiah and Owen gave to Hannah the same.  LCM of 4 and 6 is 12.

Number of pens that Hannah had more than Owen 
= 36 u - 11 u
= 25 u

25 u = 150

1 u = 150 ÷ 25 = 6

Number of pens that Josiah gave to Hannah
= 3 u
= 3 x 6
= 18

Answer(s): 18