If Cathy gives Ryan $35, the amount of money she has will be 10% of his. If Ryan gives Cathy $172, he will have the same amount of money as her. How much money did each of them have?
(a) Ryan
(b) Cathy

	Case 1		Case 2
	Ryan	Cathy	Ryan	Cathy
Before	10 u - 35	1 u + 35	5.5 u + 172	5.5 u - 172
Change	+ 35	- 35	- 172	+ 172
After	10 u	1 u	5.5 u	5.5 u

(a)
10% = ¹⁰₁₀₀ = ¹₁₀

In Case 1, when Cathy gives to Ryan, there is an internal transfer of money from Cathy to Ryan.  The total amount that both have remains the same.

In Case 2, when Ryan gives to Cathy, there is an internal transfer of money from Ryan to Cathy. The total amount that both have remains the same.

The total amount that both have in Case 1 and Case 2 are the same.

Total amount that Cathy and Ryan have
= 10 u + 1 u
= 11 u

Amount that Cathy and Ryan each has in the end for Case 2
= 11 u ÷ 2
= 5.5 u

Amount that Ryan has at first is the same in Case 1 and Case 2.

10 u - 35 = 5.5 u + 172
10 u - 5.5 u = 172 + 35
4.5 u = 207 

1 u = 207 ÷ 4.5 = 46

Amount that Ryan has
= 10 u - 35
= 10 x 46 - 35
= 460 - 35
= $425 

(b)
Amount that Cathy has
= 1 u + 35
= 1 x 46 + 35
= 46 + 35
= $81

Answer(s): (a) $425; (b) $81