Jeremy bought 90 purple and yellow balloons for a party. The number of yellow balloons was ⁴₅ times as many as the number of purple balloons. Jeremy used ¹₃ times as many yellow balloons as purple balloons and was left with twice as many yellow balloons as purple balloons. How many balloons did Jeremy have in the end?

	Yellow (1)	Purple (2)	Make b the same. (2)x2 = (3)
Before	40	50	100
Change	- 1 p	- 3 p	- 6 p
After	2 b	1 b	2 b

Number of yellow balloons that Jeremy bought at first = 4 u
Number of purple balloons that Jeremy bought at first = 5 u

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

9 u = 90

1 u = 90 ÷ 9 = 10

Number of yellow balloons at first
= 4 u
= 4 x 10
= 40

Number of purple balloons at first
= 5 u
= 5 x 10
= 50

40 - 1 p = 2 b --- (1)
50 - 3 p = 1 b --- (2)

(2)_x2
100 - 6 p = 2 b --- (3)

(3) = (1)
100 - 6 p = 40 - 1 p
6 p - 1 p = 100 - 40 
5 p = 60

1 p = 60 ÷ 5 = 12

Number of balloons that Jeremy used
= 1 p + 3 p
= 4 p
= 4 x 12
= 48

Number of balloons that Jeremy had in the end
= 90 - 48
= 42

Answer(s): 42