Luke bought 176 purple and grey balloons for a party. The number of grey balloons was ⁴₇ times as many as the number of purple balloons. Luke used ¹₃ times as many grey balloons as purple balloons and was left with twice as many grey balloons as purple balloons. How many balloons did Luke have in the end?

	Grey (1)	Purple (2)	Make b the same. (2)x2 = (3)
Before	64	112	224
Change	- 1 p	- 3 p	- 6 p
After	2 b	1 b	2 b

Number of grey balloons that Luke bought at first = 4 u
Number of purple balloons that Luke bought at first = 7 u

Total number of balloons at first
= 4 u + 7 u
= 11 u

11 u = 176

1 u = 176 ÷ 11 = 16

Number of grey balloons at first
= 4 u
= 4 x 16
= 64

Number of purple balloons at first
= 7 u
= 7 x 16
= 112

64 - 1 p = 2 b --- (1)
112 - 3 p = 1 b --- (2)

(2)_x2
224 - 6 p = 2 b --- (3)

(3) = (1)
224 - 6 p = 64 - 1 p
6 p - 1 p = 224 - 64 
5 p = 160

1 p = 160 ÷ 5 = 32

Number of balloons that Luke used
= 1 p + 3 p
= 4 p
= 4 x 32
= 128

Number of balloons that Luke had in the end
= 176 - 128
= 48

Answer(s): 48