Three women, Olivia, Gwen and Raeann went on a shopping spree. 90% of Olivia's spending was equal to ¹₅ of Gwen's spending. Raeann's spending was  60% more than Gwen's. If Gwen spent another $1026, she would spend the same amount of money as Raeann.

1. Find the ratio of Olivia's spending to Gwen's to Raeann's.
2. How much did Olivia spend?

Olivia	Gwen	Raeann
2x5	9x5
	5x9	8x9
10 u	45 u	72 u

(a)
90%= ⁹⁰₁₀₀ = ⁹₁₀
⁹₁₀ of Olivia's spending is equal to ¹₅ of Gwen's spending. Make the numerators the same.  LCM of 9 and 1 is 9.

910 of Olivia's spending = 15 of Gwen's spending
910 of Olivia's spending =1x95x9 of Gwen's spending
910 of Olivia's spending = 945 of Gwen's spending

Olivia : Gwen
10 : 45
2 : 9

Raeann's spending in percent when compared to Gwen's
= 100% + 60%
= 160%

Gwen : Raeann
100 : 160
5 : 8

Gwen's spending is the repeated identity.  Make Gwen's spending the same.  LCM of 9 and 5 is 45.

Olivia : Gwen : Raeann
10 : 45 : 72

(b)

	Olivia	Gwen	Raeann
Before	10 u	45 u	72 u
Change		+ 27 u
After	10 u	72 u	72 u

Additional amount that Gwen would have to spend to be the same as Raeann
= 72 u - 45 u
= 27 u

27 u = 1026

1 u = 1026 ÷ 27 = 38

Amount that Olivia spent
= 10 u
= 10 x 38
= $380 

Answer(s): (a) 10 : 45 : 72; (b) $380