Natalie and Hilda have some macarons each. If Natalie gives Hilda 20 macarons, Natalie will have ²₃ as many macarons as Hilda. If Natalie gives Hilda 10 macarons, they will have an equal number of macarons. How many macarons does Natalie have at first?

	Case 1		Case 2
	Natalie	Hilda	Natalie	Hilda
Before	2 u + 20	3 u - 20	2.5 u + 10	2.5 u - 10
Change	- 20	+ 20	- 10	+ 10
After	2 u	3 u	2.5 u	2.5 u

Total number of macarons that Natalie and Hilda have
= 2 u + 3 u
= 5 u 

Number of macarons that Natalie and Hilda each has in the end is the same.

Number of macarons that Natalie and Hilda each has in the end in Case 2
= 5 u ÷ 2
= 2.5 u

Number of macarons that Natalie had at first is the same in Case 1 and Case 2.

2.5 u + 10 = 2 u + 20
2.5 u - 2 u = 20 - 10
0.5 u = 10

1 u = 10 ÷ 0.5 = 20

Number of macarons that Natalie has
= 2 u + 20
= 2 x 20 + 20
= 40 + 20
= 60

Answer(s): 60