Three women, Usha, Kathy and Gabby went on a shopping spree. 90% of Usha's spending was equal to ¹₆ of Kathy's spending. Gabby's spending was  70% less than Kathy's. If Kathy spent $21546 less, she would spend the same amount of money as Gabby.

1. Find the ratio of Usha's spending to Kathy's to Gabby's.
2. How much did Gabby spend?

Usha	Kathy	Gabby
5x10	27x10
	10x27	3x27
50 u	270 u	81 u

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

910 of Usha's spending = 16 of Kathy's spending
910 of Usha's spending =1x96x9 of Kathy's spending
910 of Usha's spending = 954 of Kathy's spending

Usha : Kathy
10 : 54
5 : 27

Gabby's spending in percent when compared to Kathy's
= 100% - 70%
= 30%

Kathy : Gabby
100 : 30
10 : 3

Kathy's spending is the repeated identity.  Make Kathy's spending the same.  LCM of 27 and 10 is 270.

Usha : Kathy : Gabby
50 : 270 : 81

(b)

	Usha	Kathy	Gabby
Before	50 u	270 u	81 u
Change		- 189 u
After	50 u	81 u	81 u

Additional amount that Kathy would have to spend less to be the same as Gabby
= 270 u - 81 u
= 189 u

189 u = 21546

1 u = 21546 ÷ 189 = 114

Amount that Gabby spent
= 81 u
= 81 x 114
= $9234 

Answer(s): (a) 50 : 270 : 81; (b) $9234