Reggie, Rael and Japheth had a total of 142 pencils. The ratio of Rael's pencils to Japheth's pencils was 4 : 7 at first. Reggie and Rael each gave away ¹₂ of their pencils. Given that the three boys had 113 pencils left, how many pencils did Reggie have at first?

	Reggie	Rael	Japheth	Total
Comparing Rael and Japheth at first		4 u	7 u
Before	2 p	2x2 = 4 u	7 u	142
Change	- 1 p	-1x2 = - 2 u		- 29
After	1 p	1x2 = 2 u	7 u	113

Total number of pencils that Reggie and Rael gave away
= 142 - 113
= 29

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 4 and 2 is 4.

1 p + 2 u = 142 - 113
1 p + 2 u = 29
1 p = 29 - 2 u --- (1)

1 p + 2 u + 7 u = 113
1 p + 9 u = 113
1 p = 113 - 9 u --- (2)

(1) = (2)
29 - 2 u = 113 - 9 u
9 u - 2 u = 113 - 29
9 u - 2 u = 84
7 u = 84

1 u = 84 ÷ 7 = 12

Substitute 1 u = 12 into (1).
1 p = 29 - 2 u
1 p = 29 - 2 x 12
1 p = 29 - 24

1 p = 5

Number of pencils that Reggie had at first
= 2 p
= 2 x 5
= 10

Answer(s): 10