2/5 of Zoe's money was 12 dollars more than 2/3 of Luke's money. When Zoe gave Luke 50 dollars, both of them had the same amount of money in the end. How much did Luke have at first?
+128460 Let Z = what Zoe started with and L = what Luke started with
(2/5)Z = (2/3)L + 12 Multiply through by 15
6Z = 10L + 180
Z = (10L + 180) / 6 (1)
So
Z - 50 = L + 50 (2) sub (1) into (2)
(10L + 180) / 6 - 50 = L + 50
(5/3)L + 30 - 50 = L + 50
(5/3)L - 20 = L + 50
(5/3)L - L = 50 + 20
(2/3)L = 70
L = 70 (3/2) = $105 = what Luke started with
