There are some white and purple beads in a box. If 3 white beads are removed from the box, the total number of beads left will be 9 times the number of white beads left. If 32 purple beads are removed from the box, the total number of beads left will be 5 times the number of white beads. How many beads are there in the box?

	Case 1		Case 2
	White beads	Purple beads	White beads	Purple beads
Before	1 u + 3	8 u	1 p	4 p + 32
Change	- 3			- 32
After	1 u	8 u	1 p	4 p

Number of purple beads in the end for Case 1
= 9 u - 1 u
= 8 u

Number of white beads in the end for Case 2
= 5 p - 1 p
= 4 p

The number of white 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 + 3 = 1 p --- (1)

8 u = 4 p + 32
8 u - 32 = 4 p --- (2)

Make p the same.

(1)_x4
4 u + 12 = 4 p --- (3)

(2) = (3)
8 u - 32 = 4 u + 12
8 u - 4 u = 12 + 32
4 u = 44

1 u = 44 ÷ 4 = 11

Number of beads in the box
= (1 u + 3) + 8 u
= 9 u + 3
= (9 x 11) + 3
= 99 + 3
= 102

Answer(s): 102