Henry, Owen and Carl had a total of 136 pencils. The ratio of Owen's pencils to Carl's pencils was 4 : 7 at first. Henry and Owen each gave away ¹₂ of their pencils. Given that the three boys had 110 pencils left, how many pencils did Henry have in the end?

	Henry	Owen	Carl	Total
Comparing Owen and Carl at first		4 u	7 u
Before	2 p	2x2 = 4 u	7 u	136
Change	- 1 p	-1x2 = - 2 u		- 26
After	1 p	1x2 = 2 u	7 u	110

Total number of pencils that Henry and Owen gave away
= 136 - 110
= 26

The number of pencils that Owen had at first is repeated.  Make the number of pencils that Owen had at first the same.  LCM of 4 and 2 is 4.

1 p + 2 u = 136 - 110
1 p + 2 u = 26
1 p = 26 - 2 u --- (1)

1 p + 2 u + 7 u = 110
1 p + 9 u = 110
1 p = 110 - 9 u --- (2)

(1) = (2)
26 - 2 u = 110 - 9 u
9 u - 2 u = 110 - 26
9 u - 2 u = 84
7 u = 84

1 u = 84 ÷ 7 = 12

Substitute 1 u = 12 into (1).
1 p = 26 - 2 u
1 p = 26 - 2 x 12
1 p = 26 - 24

1 p = 2

Number of pencils that Henry had in the end
= 1 p
= 2

Answer(s): 2