Emily had $686 less than Valen. Emily spent ¹₅ of her money and Valen spent ²₃ of his money. In the end, Valen and Emily had the same amount of money left. Find the amount of money Valen had at first.

	Emily	Valen	Difference
Before	5x1 = 5 u	3x4 = 12 u	7 u
Change	- 1x1 = - 1 u	- 2x4 = - 8 u
After	4x1 = 4 u	1x4 = 4 u

Fraction of Emily's money left
= 1 - ¹₅
= ⁴₅

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

Valen and Emily had the same amount of money in the end. Make the numerators the same. LCM of 4 and 1 is 4.

⁴₅ of Emily's money = ¹₃ of Valen's money
⁴₅ of Emily's money = ⁴₁₂ of Valen's money

Amount that Emily had at first = 5 u
Amount that Valen had at first = 12 u

Amount that Emily had less than Valen had at first
= 12 u - 5 u
= 7 u

7 u = 686

1 u = 686 ÷ 7 = 98

Amount that Valen had at first
= 12 u
= 12 x 98
= $1176

Answer(s): $1176