Ken, Carl and Japheth had a total of 55 erasers. The ratio of Carl's erasers to Japheth's erasers was 8 : 3 at first. Ken and Carl each gave away ¹₂ of their erasers. Given that the three boys had 32 erasers left, how many erasers did Ken have at first?

	Ken	Carl	Japheth	Total
Comparing Carl and Japheth at first		8 u	3 u
Before	2 p	2x4 = 8 u	3 u	55
Change	- 1 p	-1x4 = - 4 u		- 23
After	1 p	1x4 = 4 u	3 u	32

Total number of erasers that Ken and Carl gave away
= 55 - 32
= 23

The number of erasers that Carl had at first is repeated.  Make the number of erasers that Carl had at first the same.  LCM of 8 and 2 is 8.

1 p + 4 u = 55 - 32
1 p + 4 u = 23
1 p = 23 - 4 u --- (1)

1 p + 4 u + 3 u = 32
1 p + 7 u = 32
1 p = 32 - 7 u --- (2)

(1) = (2)
23 - 4 u = 32 - 7 u
7 u - 4 u = 32 - 23
7 u - 4 u = 9
3 u = 9

1 u = 9 ÷ 3 = 3

Substitute 1 u = 3 into (1).
1 p = 23 - 4 u
1 p = 23 - 4 x 3
1 p = 23 - 12

1 p = 11

Number of erasers that Ken had at first
= 2 p
= 2 x 11
= 22

Answer(s): 22