Three women, Lucy, Xuan and Jane went on a shopping spree. 90% of Lucy's spending was equal to ¹₆ of Xuan's spending. Jane's spending was  50% more than Xuan's. If Xuan spent another $1728, she would spend the same amount of money as Jane.

1. Find the ratio of Lucy's spending to Xuan's to Jane's.
2. How much did Lucy spend?

Lucy	Xuan	Jane
5x2	27x2
	2x27	3x27
10 u	54 u	81 u

(a)
90%= ⁹⁰₁₀₀ = ⁹₁₀
⁹₁₀ of Lucy's spending is equal to ¹₆ of Xuan's spending. Make the numerators the same.  LCM of 9 and 1 is 9.

910 of Lucy's spending = 16 of Xuan's spending
910 of Lucy's spending =1x96x9 of Xuan's spending
910 of Lucy's spending = 954 of Xuan's spending

Lucy : Xuan
10 : 54
5 : 27

Jane's spending in percent when compared to Xuan's
= 100% + 50%
= 150%

Xuan : Jane
100 : 150
2 : 3

Xuan's spending is the repeated identity.  Make Xuan's spending the same.  LCM of 27 and 2 is 54.

Lucy : Xuan : Jane
10 : 54 : 81

(b)

	Lucy	Xuan	Jane
Before	10 u	54 u	81 u
Change		+ 27 u
After	10 u	81 u	81 u

Additional amount that Xuan would have to spend to be the same as Jane
= 81 u - 54 u
= 27 u

27 u = 1728

1 u = 1728 ÷ 27 = 64

Amount that Lucy spent
= 10 u
= 10 x 64
= $640 

Answer(s): (a) 10 : 54 : 81; (b) $640