Three women, Cathy, Anna and Gem went on a shopping spree. 30% of Cathy's spending was equal to ¹₂ of Anna's spending. Gem's spending was  70% less than Anna's. If Anna spent $1890 less, she would spend the same amount of money as Gem.

1. Find the ratio of Cathy's spending to Anna's to Gem's.
2. How much did Gem spend?

Cathy	Anna	Gem
5x10	3x10
	10x3	3x3
50 u	30 u	9 u

(a)
30%= ³⁰₁₀₀ = ³₁₀
³₁₀ of Cathy's spending is equal to ¹₂ of Anna's spending. Make the numerators the same.  LCM of 3 and 1 is 3.

310 of Cathy's spending = 12 of Anna's spending
310 of Cathy's spending =1x32x3 of Anna's spending
310 of Cathy's spending = 36 of Anna's spending

Cathy : Anna
10 : 6
5 : 3

Gem's spending in percent when compared to Anna's
= 100% - 70%
= 30%

Anna : Gem
100 : 30
10 : 3

Anna's spending is the repeated identity.  Make Anna's spending the same.  LCM of 3 and 10 is 30.

Cathy : Anna : Gem
50 : 30 : 9

(b)

	Cathy	Anna	Gem
Before	50 u	30 u	9 u
Change		- 21 u
After	50 u	9 u	9 u

Additional amount that Anna would have to spend less to be the same as Gem
= 30 u - 9 u
= 21 u

21 u = 1890

1 u = 1890 ÷ 21 = 90

Amount that Gem spent
= 9 u
= 9 x 90
= $810 

Answer(s): (a) 50 : 30 : 9; (b) $810