Vanessa and Hazel have some cupcakes each. If Vanessa gives Hazel 29 cupcakes, Vanessa will have ⁴₅ as many cupcakes as Hazel. If Vanessa gives Hazel 20 cupcakes, they will have an equal number of cupcakes. How many cupcakes does Vanessa have at first?

	Case 1		Case 2
	Vanessa	Hazel	Vanessa	Hazel
Before	4 u + 29	5 u - 29	4.5 u + 20	4.5 u - 20
Change	- 29	+ 29	- 20	+ 20
After	4 u	5 u	4.5 u	4.5 u

Total number of cupcakes that Vanessa and Hazel have
= 4 u + 5 u
= 9 u 

Number of cupcakes that Vanessa and Hazel each has in the end is the same.

Number of cupcakes that Vanessa and Hazel each has in the end in Case 2
= 9 u ÷ 2
= 4.5 u

Number of cupcakes that Vanessa had at first is the same in Case 1 and Case 2.

4.5 u + 20 = 4 u + 29
4.5 u - 4 u = 29 - 20
0.5 u = 9

1 u = 9 ÷ 0.5 = 18

Number of cupcakes that Vanessa has
= 4 u + 29
= 4 x 18 + 29
= 72 + 29
= 101

Answer(s): 101