Archie and Carl had 78 beads in total. Archie had 20 beads less than Carl at first. After Archie and Carl played a game with Rachel, Archie had ¹₆ as many beads as Carl. If Archie lost as many beads as Carl to Rachel, find the number of beads that Carl lost to Rachel.

	Archie	Carl	Difference
Before	1 u	1 u + 20	20
Change	- ?	- ?
After	1 p	6 p	5 p

Total number of beads at first
= 1 u + 1 u + 20 
= 2 u + 20

2 u + 20 = 78
2 u = 78 - 20
2 u = 58

1 u = 58 ÷ 2 = 29

Since Archie lost the same number of beads as Carl, the difference in the number of beads between Archie and Carl at first and in the end remains the same.

Difference in the number of beads in the end
= 6 p - 1 p
= 5 p

5 p = 20

1 p = 20 ÷ 5 = 4

Number of beads that Archie lost to Rachel
= 1 u - 1 p
= 29 - 1 x 4
= 29 - 4
= 25

Since Carl lost the same number of beads as Archie, the number of beads that each lost to Rachel is the same.

Number of beads that Carl lost = 25

Answer(s): 25