Nick, Ivan and Japheth had a total of 83 erasers. The ratio of Ivan's erasers to Japheth's erasers was 8 : 7 at first. Nick and Ivan each gave away ¹₂ of their erasers. Given that the three boys had 59 erasers left, how many erasers did Nick have in the end?

	Nick	Ivan	Japheth	Total
Comparing Ivan and Japheth at first		8 u	7 u
Before	2 p	2x4 = 8 u	7 u	83
Change	- 1 p	-1x4 = - 4 u		- 24
After	1 p	1x4 = 4 u	7 u	59

Total number of erasers that Nick and Ivan gave away
= 83 - 59
= 24

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

1 p + 4 u = 83 - 59
1 p + 4 u = 24
1 p = 24 - 4 u --- (1)

1 p + 4 u + 7 u = 59
1 p + 11 u = 59
1 p = 59 - 11 u --- (2)

(1) = (2)
24 - 4 u = 59 - 11 u
11 u - 4 u = 59 - 24
11 u - 4 u = 35
7 u = 35

1 u = 35 ÷ 7 = 5

Substitute 1 u = 5 into (1).
1 p = 24 - 4 u
1 p = 24 - 4 x 5
1 p = 24 - 20

1 p = 4

Number of erasers that Nick had in the end
= 1 p
= 4

Answer(s): 4