Level d:

rollie 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?

explain how you arrived at your solution.

If X is the starting weight, total pounds lost in three months is

(0.132X + 2.37)lbs

Step-by-step explanation:

Let the starting weight be X(lb)

Let the target ending weight be Y(lb)

Since Rollie ended with 3(lbs) above goal weight, final weight is Y+3(lb)

Each month, Rollie lost 1/3 of the difference between current weight and goal weight, hence

In the first month, Rollie lost 1/3 of (X - Y)

but since Rollie ended up with 3lbs above target ending weight, the actual weight loss is 1/3(X - Y) + 3

Hence  by the end of the first month, Rollie's new weight is X - (1/3(X - Y)) + 3

which is equal to

X - 1/3X + 1/3Y - 3.

= 2/3X + 1/3Y - 3

By the second month, Rollie's starting weight is 2/3X + 1/3Y - 3(lb)

while her target ending weight still remains (Y)

but the actual ending weight is Y + 3

hence in the second month, Rollie lost 1/3(2/3X + 1/3Y - 3 - Y) + 3

= 2/9X + 1/9Y - 1 - 1/3Y + 3

=  2/9X - 2/9Y + 2

hence by the end of the second month, Rollie's new weight is

(2/3X + 1/3Y - 3) - (2/9X - 2/9Y + 2)

= 2/3X - 2/9X + 1/3Y + 2/9Y - 3 - 2

= 4/9X + 5/9Y - 5  

By the third month, Rollie's starting weight is 4/9X + 5/9Y - 5(lb)

while her target ending weight still remains (Y)

but the actual ending weight is Y + 3

hence in the third month, Rollie lost 1/3(4/9X + 5/9Y - 5 - Y) + 3

= 4/27X + 5/27Y - 5/3 - 1/3Y + 3

= 4/27X - 4/27Y + 4/3 eqn (*)

hence by the end of the third month, Rollie's new weight is

(4/9X + 5/9Y - 5) - (4/27X - 4/27Y + 4/3)

= 4/9X - 4/27X + 5/9Y + 4/27Y - 5 - 4/3

= 8/27X - 19/27Y - 19/3

Hence in three months, Rollie's new weight is 8/27X - 19/27Y - 19/3

To ascertain how much weight Rollie lost in three months, there is need to equate the estimated ending weight to the final weight (Y + 3)

Therefore:

8/27X - 19/27Y - 19/3 = Y + 3

this implies that

8/27X - 19/27Y - Y - 19/3 - 3 = 0

8/27X - 36/27Y -28/3 = 0

Since Y is the target ending weight

the ending weight is generated by solving the equation with respect to Y

36/27Y = 8/27X - 28/3

multiply through by 27/36

Y = 2/9X - 7

Hence weight lost in three months is generated by substituting for Y = 2/9X - 7  in eqn (*)

Since eqn * is 4/27X - 4/27Y + 4/3

the lbs lost in three months is

4/27X - 4/27(2/9X - 7 ) + 4/3

= 4/27X - 8/243X + 28/27 + 4/3

= (32/243)X + 64/27

which in decimal is

(0.132X + 2.37)lbs

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