Japheth and Daniel had 84 beads in total. Japheth had 16 beads less than Daniel at first. After Japheth and Daniel played a game with Kathy, Japheth had ¹₅ as many beads as Daniel. If Japheth lost as many beads as Daniel to Kathy, find the number of beads that Daniel lost to Kathy.

	Japheth	Daniel	Difference
Before	1 u	1 u + 16	16
Change	- ?	- ?
After	1 p	5 p	4 p

Total number of beads at first
= 1 u + 1 u + 16 
= 2 u + 16

2 u + 16 = 84
2 u = 84 - 16
2 u = 68

1 u = 68 ÷ 2 = 34

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

Difference in the number of beads in the end
= 5 p - 1 p
= 4 p

4 p = 16

1 p = 16 ÷ 4 = 4

Number of beads that Japheth lost to Kathy
= 1 u - 1 p
= 34 - 1 x 4
= 34 - 4
= 30

Since Daniel lost the same number of beads as Japheth, the number of beads that each lost to Kathy is the same.

Number of beads that Daniel lost = 30

Answer(s): 30