Japheth, Ryan and Sean had a total of 89 coins. The ratio of Ryan's coins to Sean's coins was 4 : 3 at first. Japheth and Ryan each gave away ¹₂ of their coins. Given that the three boys had 61 coins left, how many coins did Japheth have in the end?

	Japheth	Ryan	Sean	Total
Comparing Ryan and Sean at first		4 u	3 u
Before	2 p	2x2 = 4 u	3 u	89
Change	- 1 p	-1x2 = - 2 u		- 28
After	1 p	1x2 = 2 u	3 u	61

Total number of coins that Japheth and Ryan gave away
= 89 - 61
= 28

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

1 p + 2 u = 89 - 61
1 p + 2 u = 28
1 p = 28 - 2 u --- (1)

1 p + 2 u + 3 u = 61
1 p + 5 u = 61
1 p = 61 - 5 u --- (2)

(1) = (2)
28 - 2 u = 61 - 5 u
5 u - 2 u = 61 - 28
5 u - 2 u = 33
3 u = 33

1 u = 33 ÷ 3 = 11

Substitute 1 u = 11 into (1).
1 p = 28 - 2 u
1 p = 28 - 2 x 11
1 p = 28 - 22

1 p = 6

Number of coins that Japheth had in the end
= 1 p
= 6

Answer(s): 6