Three women, Elyse, Raeann and Natalie went on a shopping spree. 30% of Elyse's spending was equal to ¹₆ of Raeann's spending. Natalie's spending was  70% less than Raeann's. If Raeann spent $6300 less, she would spend the same amount of money as Natalie.

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

Elyse	Raeann	Natalie
5x10	9x10
	10x9	3x9
50 u	90 u	27 u

(a)
30%= ³⁰₁₀₀ = ³₁₀
³₁₀ of Elyse's spending is equal to ¹₆ of Raeann's spending. Make the numerators the same.  LCM of 3 and 1 is 3.

310 of Elyse's spending = 16 of Raeann's spending
310 of Elyse's spending =1x36x3 of Raeann's spending
310 of Elyse's spending = 318 of Raeann's spending

Elyse : Raeann
10 : 18
5 : 9

Natalie's spending in percent when compared to Raeann's
= 100% - 70%
= 30%

Raeann : Natalie
100 : 30
10 : 3

Raeann's spending is the repeated identity.  Make Raeann's spending the same.  LCM of 9 and 10 is 90.

Elyse : Raeann : Natalie
50 : 90 : 27

(b)

	Elyse	Raeann	Natalie
Before	50 u	90 u	27 u
Change		- 63 u
After	50 u	27 u	27 u

Additional amount that Raeann would have to spend less to be the same as Natalie
= 90 u - 27 u
= 63 u

63 u = 6300

1 u = 6300 ÷ 63 = 100

Amount that Natalie spent
= 27 u
= 27 x 100
= $2700 

Answer(s): (a) 50 : 90 : 27; (b) $2700