Three women, Rachel, Vanessa and Xylia went on a shopping spree. 70% of Rachel's spending was equal to ¹₂ of Vanessa's spending. Xylia's spending was  40% more than Vanessa's. If Vanessa spent another $896, she would spend the same amount of money as Xylia.

1. Find the ratio of Rachel's spending to Vanessa's to Xylia's.
2. How much did Rachel spend?

Rachel	Vanessa	Xylia
5x5	7x5
	5x7	7x7
25 u	35 u	49 u

(a)
70%= ⁷⁰₁₀₀ = ⁷₁₀
⁷₁₀ of Rachel's spending is equal to ¹₂ of Vanessa's spending. Make the numerators the same.  LCM of 7 and 1 is 7.

710 of Rachel's spending = 12 of Vanessa's spending
710 of Rachel's spending =1x72x7 of Vanessa's spending
710 of Rachel's spending = 714 of Vanessa's spending

Rachel : Vanessa
10 : 14
5 : 7

Xylia's spending in percent when compared to Vanessa's
= 100% + 40%
= 140%

Vanessa : Xylia
100 : 140
5 : 7

Vanessa's spending is the repeated identity.  Make Vanessa's spending the same.  LCM of 7 and 5 is 35.

Rachel : Vanessa : Xylia
25 : 35 : 49

(b)

	Rachel	Vanessa	Xylia
Before	25 u	35 u	49 u
Change		+ 14 u
After	25 u	49 u	49 u

Additional amount that Vanessa would have to spend to be the same as Xylia
= 49 u - 35 u
= 14 u

14 u = 896

1 u = 896 ÷ 14 = 64

Amount that Rachel spent
= 25 u
= 25 x 64
= $1600 

Answer(s): (a) 25 : 35 : 49; (b) $1600