There are some grey and purple beads in a container. If 8 grey beads are removed from the container, the total number of beads left will be 8 times the number of grey beads left. If 8 purple beads are removed from the container, the total number of beads left will be 4 times the number of grey beads. How many beads are there in the container?

	Case 1		Case 2
	Grey beads	Purple beads	Grey beads	Purple beads
Before	1 u + 8	7 u	1 p	3 p + 8
Change	- 8			- 8
After	1 u	7 u	1 p	3 p

Number of purple beads in the end for Case 1
= 8 u - 1 u
= 7 u

Number of grey beads in the end for Case 2
= 4 p - 1 p
= 3 p

The number of grey beads at first is the same for Case 1 and Case 2.
The number of purple beads at first is also the same for Case 1 and Case 2.

1 u + 8 = 1 p --- (1)

7 u = 3 p + 8
7 u - 8 = 3 p --- (2)

Make p the same.

(1)_x3
3 u + 24 = 3 p --- (3)

(2) = (3)
7 u - 8 = 3 u + 24
7 u - 3 u = 24 + 8
4 u = 32

1 u = 32 ÷ 4 = 8

Number of beads in the container
= (1 u + 8) + 7 u
= 8 u + 8
= (8 x 8) + 8
= 64 + 8
= 72

Answer(s): 72