Gavin had 50% more pencils than Rael. Avery had 20% fewer pencils than Gavin. Gavin and Rael gave Avery some pencils in the ratio of 4 : 5. As a result, Avery had 50% more pencils than before. Given that Rael had 204 less pencils than Avery in the end, how many pencils did Rael give to Avery?

Gavin	Rael	Avery
3x5	2x5
5x3		4x3
15	10	12

100% + 50% = 150%
150% = ¹⁵⁰₁₀₀ = ³₂

Gavin : Rael = 3 : 2

100% - 20% = 80%
80% = ⁸⁰₁₀₀ = ⁴₅

Gavin : Avery = 5 : 4

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

	Gavin	Rael	Avery
Before	15x3 =  45 u	10x3 = 30 u	12x3 = 36 u
Change			+ 6x3 = + 18 u
Comparing the changes in the number of pencils	- 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 pencils that Gavin and Rael gave to Avery is repeated.  Make the total number of pencils that Gavin and Rael gave to Avery the same.  LCM of 9 and 6 is 18.

Number of pencils that Rael had less than Avery 
= 54 u - 20 u
= 34 u

34 u = 204

1 u = 204 ÷ 34 = 6

Number of pencils that Rael gave to Avery
= 10 u
= 10 x 6
= 60

Answer(s): 60