Cole, Harry and Albert had a total of 59 beads. The ratio of Harry's beads to Albert's beads was 2 : 5 at first. Cole and Harry each gave away ¹₂ of their beads. Given that the three boys had 42 beads left, how many beads did Cole have at first?

	Cole	Harry	Albert	Total
Comparing Harry and Albert at first		2 u	5 u
Before	2 p	2x1 = 2 u	5 u	59
Change	- 1 p	-1x1 = - 1 u		- 17
After	1 p	1x1 = 1 u	5 u	42

Total number of beads that Cole and Harry gave away
= 59 - 42
= 17

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

1 p + 1 u = 59 - 42
1 p + 1 u = 17
1 p = 17 - 1 u --- (1)

1 p + 1 u + 5 u = 42
1 p + 6 u = 42
1 p = 42 - 6 u --- (2)

(1) = (2)
17 - 1 u = 42 - 6 u
6 u - 1 u = 42 - 17
6 u - 1 u = 25
5 u = 25

1 u = 25 ÷ 5 = 5

Substitute 1 u = 5 into (1).
1 p = 17 - 1 u
1 p = 17 - 1 x 5
1 p = 17 - 5

1 p = 12

Number of beads that Cole had at first
= 2 p
= 2 x 12
= 24

Answer(s): 24