Three women, Gwen, Jen and Abi went on a shopping spree. 30% of Gwen's spending was equal to ¹₆ of Jen's spending. Abi's spending was  40% less than Jen's. If Jen spent $414 less, she would spend the same amount of money as Abi.

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

Gwen	Jen	Abi
5x5	9x5
	5x9	3x9
25 u	45 u	27 u

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

310 of Gwen's spending = 16 of Jen's spending
310 of Gwen's spending =1x36x3 of Jen's spending
310 of Gwen's spending = 318 of Jen's spending

Gwen : Jen
10 : 18
5 : 9

Abi's spending in percent when compared to Jen's
= 100% - 40%
= 60%

Jen : Abi
100 : 60
5 : 3

Jen's spending is the repeated identity.  Make Jen's spending the same.  LCM of 9 and 5 is 45.

Gwen : Jen : Abi
25 : 45 : 27

(b)

	Gwen	Jen	Abi
Before	25 u	45 u	27 u
Change		- 18 u
After	25 u	27 u	27 u

Additional amount that Jen would have to spend less to be the same as Abi
= 45 u - 27 u
= 18 u

18 u = 414

1 u = 414 ÷ 18 = 23

Amount that Abi spent
= 27 u
= 27 x 23
= $621 

Answer(s): (a) 25 : 45 : 27; (b) $621