Ethan bought 126 purple and silver balloons for a party. The number of silver balloons was ⁴₅ times as many as the number of purple balloons. Ethan used ¹₄ times as many silver balloons as purple balloons and was left with thrice as many silver balloons as purple balloons. How many balloons did Ethan have in the end?

	Silver (1)	Purple (2)	Make b the same. (2)x3 = (3)
Before	56	70	210
Change	- 1 p	- 4 p	- 12 p
After	3 b	1 b	3 b

Number of silver balloons that Ethan bought at first = 4 u
Number of purple balloons that Ethan bought at first = 5 u

Total number of balloons at first
= 4 u + 5 u
= 9 u

9 u = 126

1 u = 126 ÷ 9 = 14

Number of silver balloons at first
= 4 u
= 4 x 14
= 56

Number of purple balloons at first
= 5 u
= 5 x 14
= 70

56 - 1 p = 3 b --- (1)
70 - 4 p = 1 b --- (2)

(2)_x3
210 - 12 p = 3 b --- (3)

(3) = (1)
210 - 12 p = 56 - 1 p
12 p - 1 p = 210 - 56 
11 p = 154

1 p = 154 ÷ 11 = 14

Number of balloons that Ethan used
= 1 p + 4 p
= 5 p
= 5 x 14
= 70

Number of balloons that Ethan had in the end
= 126 - 70
= 56

Answer(s): 56