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

Joseph	Grace	Risa
6x5	5x5
5x6		3x6
30	25	18

100% + 20% = 120%
120% = ¹²⁰₁₀₀ = ⁶₅

Joseph : Grace = 6 : 5

100% - 40% = 60%
60% = ⁶⁰₁₀₀ = ³₅

Joseph : Risa = 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	Grace	Risa
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 Grace gave to Risa is repeated.  Make the total number of pencils that Joseph and Grace gave to Risa the same.  LCM of 6 and 9 is 18.

Number of pencils that Grace had less than Risa 
= 54 u - 35 u
= 19 u

19 u = 57

1 u = 57 ÷ 19 = 3

Number of pencils that Grace gave to Risa
= 15 u
= 15 x 3
= 45

Answer(s): 45