Ryan, Ivan and Ahmad had a total of 58 beads. The ratio of Ivan's beads to Ahmad's beads was 2 : 9 at first. Ryan and Ivan each gave away ¹₂ of their beads. Given that the three boys had 47 beads left, how many beads did Ryan have in the end?

	Ryan	Ivan	Ahmad	Total
Comparing Ivan and Ahmad at first		2 u	9 u
Before	2 p	2x1 = 2 u	9 u	58
Change	- 1 p	-1x1 = - 1 u		- 11
After	1 p	1x1 = 1 u	9 u	47

Total number of beads that Ryan and Ivan gave away
= 58 - 47
= 11

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

1 p + 1 u = 58 - 47
1 p + 1 u = 11
1 p = 11 - 1 u --- (1)

1 p + 1 u + 9 u = 47
1 p + 10 u = 47
1 p = 47 - 10 u --- (2)

(1) = (2)
11 - 1 u = 47 - 10 u
10 u - 1 u = 47 - 11
10 u - 1 u = 36
9 u = 36

1 u = 36 ÷ 9 = 4

Substitute 1 u = 4 into (1).
1 p = 11 - 1 u
1 p = 11 - 1 x 4
1 p = 11 - 4

1 p = 7

Number of beads that Ryan had in the end
= 1 p
= 7

Answer(s): 7