Joseph had 20% more pencils than Harper. Hazel had 40% fewer pencils than Joseph. Joseph and Harper gave Hazel some pencils in the ratio of 1 : 5. As a result, Hazel had 50% more pencils than before. Given that Hazel had 57 more pencils than Harper in the end, how many pencils did Joseph give to Hazel?

Joseph	Harper	Hazel
6x5	5x5
5x6		3x6
30	25	18

100% + 20% = 120%
120% = ¹²⁰₁₀₀ = ⁶₅

Joseph : Harper = 6 : 5

100% - 40% = 60%
60% = ⁶⁰₁₀₀ = ³₅

Joseph : Hazel = 5 : 3

The number of pencils that Joseph had at first is repeated. Make the number of pencils that Joseph had at first the same.  LCM of 6 and 5 is 30.

	Joseph	Harper	Hazel
Before	30x2 =  60 u	25x2 = 50 u	18x2 = 36 u
Change			+ 9x2 = + 18 u
Comparing the changes in the number of pencils	- 1x3 = - 3 u	- 5x3 = - 15 u	+ 6x3 = + 18 u
After	57 u	35 u	27x2 = 54 u

50% x 18 
= ⁵⁰₁₀₀ x 18 
= 9 

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

Number of pencils that Hazel had more than Harper 
= 54 u - 35 u
= 19 u

19 u = 57

1 u = 57 ÷ 19 = 3

Number of pencils that Joseph gave to Hazel
= 3 u
= 3 x 3
= 9

Answer(s): 9