If Diana gives Luke $23, the amount of money she has will be 10% of his. If Luke gives Diana $85, he will have the same amount of money as her. How much money did each of them have?
(a) Luke
(b) Diana

	Case 1		Case 2
	Luke	Diana	Luke	Diana
Before	10 u - 23	1 u + 23	5.5 u + 85	5.5 u - 85
Change	+ 23	- 23	- 85	+ 85
After	10 u	1 u	5.5 u	5.5 u

(a)
10% = ¹⁰₁₀₀ = ¹₁₀

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

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

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

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

10 u - 23 = 5.5 u + 85
10 u - 5.5 u = 85 + 23
4.5 u = 108 

1 u = 108 ÷ 4.5 = 24

Amount that Luke has
= 10 u - 23
= 10 x 24 - 23
= 240 - 23
= $217 

(b)
Amount that Diana has
= 1 u + 23
= 1 x 24 + 23
= 24 + 23
= $47

Answer(s): (a) $217; (b) $47