Kathy, Jane and Emily had a sum of money. The amount of money Kathy had was ⁴₅ of the amount of money Jane had. The ratio of the amount of money Jane had to the total amount of money Emily and Jane had was 4: 9. After Emily gave $100 to Jane, she had as much money as Jane. How much more money did Emily have than Kathy in the end?

Kathy	Jane	Emily
4x4 = 16 u	5x4 = 20 u
	4x5 = 20 u	5x5 = 25 u
16 u	20 u	25 u

The amount that Jane had is the repeated identity. Make the amount that Jane had the same.  LCM of 5 and 4 is 20.

	Kathy	Jane	Emily
Before	16 u	20 u	25 u
Change	No change	+ 100	- 100
After	16 u	20 u + 100	25 u - 100

The amount that Emily and Jane had in the end is the same.

25 u - 100 = 20 u + 100
25 u - 20 u = 100 + 100 
5 u = 200

1 u = 200 ÷ 5 = 40

Amount that Emily had more than Kathy in the end
= 25 u - 100 - 16 u
= 9 u - 100
= 9 x 40 - 100
= 360 - 100
= $260

Answer(s): $260