There are some yellow and purple balls in a box. If 8 yellow balls are removed from the box, the total number of balls left will be 9 times the number of yellow balls left. If 12 purple balls are removed from the box, the total number of balls left will be 5 times the number of yellow balls. How many balls are there in the box?

	Case 1		Case 2
	Yellow balls	Purple balls	Yellow balls	Purple balls
Before	1 u + 8	8 u	1 p	4 p + 12
Change	- 8			- 12
After	1 u	8 u	1 p	4 p

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

Number of yellow balls in the end for Case 2
= 5 p - 1 p
= 4 p

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

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

8 u = 4 p + 12
8 u - 12 = 4 p --- (2)

Make p the same.

(1)_x4
4 u + 32 = 4 p --- (3)

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

1 u = 44 ÷ 4 = 11

Number of balls in the box
= (1 u + 8) + 8 u
= 9 u + 8
= (9 x 11) + 8
= 99 + 8
= 107

Answer(s): 107