Natalie, Xylia and Emily had 630 buttons. Xylia won some of the buttons from Natalie and as a result, Xylia's buttons increased by 20%. Emily then won some buttons from Xylia and Emily's buttons increased by 40%. Finally, Emily lost some of her buttons to Natalie and Natalie's buttons increased by 25%. In the end, they realised that they each had an equal number of buttons. How many percent less did Natalie have in the end than what she had at first? Correct your answer to 1 decimal place.

Natalie	Xylia	Emily
630
	5 u
- 1 u	+ 1 u
	6 u
		5 p
	- 2 p	+ 2 p
		7 p
4 boxes
+ 1 box		- 1 box
5 boxes
1 group	1 group	1 group

20% = ²⁰₁₀₀ = ¹₅
40% = ⁴⁰₁₀₀ = ²₅
25% = ²⁵₁₀₀ = ¹₄

Working backwards.

3 groups = 630

1 group = 630 ÷ 3 = 210

1 group = 5 boxes
5 boxes = 210

1 box = 210 ÷ 5 = 42

1 group + 1 box = 7 p
210 + 42 = 7 p
7 p = 252

1 p = 252 ÷ 7 = 36

2 p = 2 x 36 = 72

1 group + 2 p = 6 u
210 + 72 = 6 u
6 u = 282

1 u = 282 ÷ 6 = 47

Number of buttons that Natalie had at first
= 4 boxes + 1 u
= (4 x 42) + 47
= 168 + 47
= 215

Percent that Natalie had less in the end than at first
= ^{215 - 210}₂₁₅ x 100%
≈ 2.3%

Answer(s): 2.3%