Losing Weight Challenge

Question

1. Reuben was successful in losing weight. He had a goal weight in mind. He went on a diet for three months. Each month, he would lose one-third of the difference between his current weight and his goal weight and an additional three pounds. At the end of three months, he was just 3 pounds over his goal weight. How many pounds did he lose in those three months?

Answer 1

Let W_k=weight after month k, and W₀=starting weight, G = goal weight
W_k+1=W_k-1/3(W_k-G)=(2/3)W_k+(1/3)G

W₁=(2/3)W₀+(1/3)G

W₂=(2/3)W₁+(1/3)G=(2/3)((2/3)W₀+(1/3)G)+(1/3)G=(4/9)W₀+(5/9)G

W₃=(2/3)W₂+(1/3)G=(2/3)((4/9)W₀+(5/9)G)+(1/3)G=(8/27)W₀+(19/27)G

(8/27)W₀+(19/27)G=G+3

(8/27)W₀=(8/27)G+3

W₀=G+81/8

Weight lost was 3 pounds short of goal.  81/8=10 1/8 He lost 7 1/8 pounds.