Let W be Rollie's initial weight, and let G stand for the "goal" weight.

Weight at beginning of first month: W

Weight lost during the first month: (1/3)(W - G) + 3

Weight at end of the first month: W - [(1/3)W - (1/3)G + 3] which you can simplify to:

(2/3)W + (1/3)G - 3 which again, is the weight at the end of the first month. Carry out similar steps for the second and third months. There's a lot of algebra, arithmetic, and fractions, so to minimize errors if you're doing this by hand (rather than with a calculator/computer which handles the algebra) I suggest you write each step separately:

Weight at beginning of second month (which is the same as weight at the end of the first month); How much weight is lost during the second month, remembering to take the "(1/3) of the difference" and simplify and then to add 3 pounds more; and weight at the end of the second month, remembering you're now subtracting. Similarly for the third month. You'll get an expression with W's and G's and fractions with 27 in the denominator.

But you also know another name for Rollie's weight at the end of three months: it is G+3, that is 3 pounds over the idea (or goal) weight. So you can set those two expressions equal. Great. Now you have one equation with two unknowns (unless in my haste I made an arithmetic mistake; I got "(8/27)W + (19/27)G - (19/3) = G+3"). So if the starting weight isn't given and the end weight isn't given, you need some other equation to be the second equation.

I would have re-checked my work before posting here but a quick search suggests this weight problem is from a pdf by the Noyce Foundation, and same PDF speaks about "In level E, students are asked to find the exact theoretical chance knowing three conditional probable events expressed as..." which makes me wonder, for two reasons, whether this document has some mathematical typos if not errors. Because one speaks of "conditional probabilities" not "conditional probable events" and secondly, because that other problem, E, does not give conditional probabilities, like "the probability of being male given you were born in North America" would be a conditional probability. But it gives just "the probability of..male" and other non-conditional probabilities. So there are some definite typos (misprints) if not outright errors in problem E...so I wonder if the same is true for problem D, the weight problem.

Maybe they mean for the teacher to make up a problem in which W or G is known, and otherwise following the pattern of the problem in question? Their suggesting "guess-and-check" as a method sounds like it's one of those problems ("He started at 200 pounds. He lost 1/3 of the difference to...plus an additional 3 pounds") they might have in mind. Please post here if you have update or clarification.

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