If Rachel gives Ryan $22, the amount of money she has will be 10% of his. If Ryan gives Rachel $113, he will have the same amount of money as her. How much money did each of them have?
(a) Ryan
(b) Rachel

	Case 1		Case 2
	Ryan	Rachel	Ryan	Rachel
Before	10 u - 22	1 u + 22	5.5 u + 113	5.5 u - 113
Change	+ 22	- 22	- 113	+ 113
After	10 u	1 u	5.5 u	5.5 u

(a)
10% = ¹⁰₁₀₀ = ¹₁₀

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

In Case 2, when Ryan gives to Rachel, there is an internal transfer of money from Ryan to Rachel. 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 Rachel and Ryan have
= 10 u + 1 u
= 11 u

Amount that Rachel 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 - 22 = 5.5 u + 113
10 u - 5.5 u = 113 + 22
4.5 u = 135 

1 u = 135 ÷ 4.5 = 30

Amount that Ryan has
= 10 u - 22
= 10 x 30 - 22
= 300 - 22
= $278 

(b)
Amount that Rachel has
= 1 u + 22
= 1 x 30 + 22
= 30 + 22
= $52

Answer(s): (a) $278; (b) $52