Natalie had $560 less than Ethan. Natalie spent ¹₃ of her money and Ethan spent ⁴₇ of his money. In the end, Ethan and Natalie had the same amount of money left. Find the amount of money Ethan had at first.

	Natalie	Ethan	Difference
Before	3x3 = 9 u	7x2 = 14 u	5 u
Change	- 1x3 = - 3 u	- 4x2 = - 8 u
After	2x3 = 6 u	3x2 = 6 u

Fraction of Natalie's money left
= 1 - ¹₃
= ²₃

Fraction of Ethan's money left
= 1 - ⁴₇
= ³₇

Ethan and Natalie had the same amount of money in the end. Make the numerators the same. LCM of 2 and 3 is 6.

²₃ of Natalie's money = ³₇ of Ethan's money
⁶₉ of Natalie's money = ⁶₁₄ of Ethan's money

Amount that Natalie had at first = 9 u
Amount that Ethan had at first = 14 u

Amount that Natalie had less than Ethan had at first
= 14 u - 9 u
= 5 u

5 u = 560

1 u = 560 ÷ 5 = 112

Amount that Ethan had at first
= 14 u
= 14 x 112
= $1568

Answer(s): $1568