Perry, Japheth and Ben had a total of 110 coins. The ratio of Japheth's coins to Ben's coins was 4 : 9 at first. Perry and Japheth each gave away ¹₂ of their coins. Given that the three boys had 91 coins left, how many coins did Perry have at first?

	Perry	Japheth	Ben	Total
Comparing Japheth and Ben at first		4 u	9 u
Before	2 p	2x2 = 4 u	9 u	110
Change	- 1 p	-1x2 = - 2 u		- 19
After	1 p	1x2 = 2 u	9 u	91

Total number of coins that Perry and Japheth gave away
= 110 - 91
= 19

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

1 p + 2 u = 110 - 91
1 p + 2 u = 19
1 p = 19 - 2 u --- (1)

1 p + 2 u + 9 u = 91
1 p + 11 u = 91
1 p = 91 - 11 u --- (2)

(1) = (2)
19 - 2 u = 91 - 11 u
11 u - 2 u = 91 - 19
11 u - 2 u = 72
9 u = 72

1 u = 72 ÷ 9 = 8

Substitute 1 u = 8 into (1).
1 p = 19 - 2 u
1 p = 19 - 2 x 8
1 p = 19 - 16

1 p = 3

Number of coins that Perry had at first
= 2 p
= 2 x 3
= 6

Answer(s): 6