Three women, Gwen, Gem and Esther went on a shopping spree. 70% of Gwen's spending was equal to ¹₆ of Gem's spending. Esther's spending was  40% less than Gem's. If Gem spent $4452 less, she would spend the same amount of money as Esther.

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

Gwen	Gem	Esther
5x5	21x5
	5x21	3x21
25 u	105 u	63 u

(a)
70%= ⁷⁰₁₀₀ = ⁷₁₀
⁷₁₀ of Gwen's spending is equal to ¹₆ of Gem's spending. Make the numerators the same.  LCM of 7 and 1 is 7.

710 of Gwen's spending = 16 of Gem's spending
710 of Gwen's spending =1x76x7 of Gem's spending
710 of Gwen's spending = 742 of Gem's spending

Gwen : Gem
10 : 42
5 : 21

Esther's spending in percent when compared to Gem's
= 100% - 40%
= 60%

Gem : Esther
100 : 60
5 : 3

Gem's spending is the repeated identity.  Make Gem's spending the same.  LCM of 21 and 5 is 105.

Gwen : Gem : Esther
25 : 105 : 63

(b)

	Gwen	Gem	Esther
Before	25 u	105 u	63 u
Change		- 42 u
After	25 u	63 u	63 u

Additional amount that Gem would have to spend less to be the same as Esther
= 105 u - 63 u
= 42 u

42 u = 4452

1 u = 4452 ÷ 42 = 106

Amount that Esther spent
= 63 u
= 63 x 106
= $6678 

Answer(s): (a) 25 : 105 : 63; (b) $6678